- Email: [email protected]
The role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks
International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
The role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks Shuang Luo a, Zhihong Zhao a,n, Huan Peng a, Hai Pu b a b
Department of Civil Engineering, Tsinghua University, Beijing 100084, China State Key Laboratory for Geomchanics & Deep Underground Engineering, China University of Mining & Technology, Xuzhou, Jiangsu 221116, China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 12 November 2015 Received in revised form 13 April 2016 Accepted 9 May 2016
Rock fractures are major conduits for fluid flow in fractured rocks, and the convective heat transfer between rock fracture surfaces and circulating fluid is a critical issue in heat recovery in fractured rocks. It has been demonstrated that fracture surface roughness has a significant influence on the mechanical, hydraulic, thermal and transport behavior of single fractures. This study aimed to assess the effects of local surface roughness of fractures on fluid flow and heat transfer processes at the macroscopic scale of fracture networks. Two distributions of Joint Roughness Coefficient (JRC) were determined based on the JRC data in Oskarshamn/Forsmark, Sweden and Bakhtiary, Iran. Two empirical models relating hydraulic apertures to mechanical apertures were considered. A total of ninety-one realizations that considered different JRC distributions and empirical models of mechanical-hydraulic apertures were studied. The results show that fracture surface roughness can affect the fluid flow and heat transfer processes in fracture networks to various extents, mainly depending on the empirical models of mechanical-hydraulic apertures. In other words, the role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks is critical, when using a model of mechanical-hydraulic apertures that predicts significant reduced hydraulic apertures. Discrete fracture networks models with the normal distribution of JRC are less permeable than those with the lognormal distribution of JRC, using the fitting parameters of in-situ JRC data. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Rock fracture Surface roughness Fluid flow Heat transfer Discrete fracture network model
1. Introduction A rock fracture consists of two rough surfaces that are in contact with each other at some locations, but separated at others. Roughness refers to the local departures from planarity,1 and a number of parameters have been proposed to characterize the fracture surface roughness, e.g., Joint Roughness Coefficient (JRC),2 Z2,3 the ratio of standard deviation ( σE ) of the varying aperture over mechanical aperture ( E ) ( σE /E ),4 and fractal dimension.5 Grasselli and his co-workers proposed a mathema* /(C + 1) to parameterize fracture surface tical expression of θmax roughness, which can be used for three-dimensional surface topography1 or two-dimensional profiles.6 The regressive relationships between these roughness parameters have also been extensively investigated .6–10 Fractures in rock masses have a controlling influence on the mechanical behavior of rock masses, since they provide planes of weakness on which further displacement can more readily occur. n
Corresponding author. E-mail address: [email protected] (Z. Zhao).
http://dx.doi.org/10.1016/j.ijrmms.2016.05.006 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
Fractures also often provide major conduits through which groundwater can flow. It has been demonstrated that fracture surface roughness has a significant influence on the mechanical, hydraulic, thermal and transport behavior of single fractures.11 Discrete fracture network (DFN) models have been widely used to simulate the coupled thermal-hydrological-mechanical-chemical (THMC) processes on the field scale of fractured rock mass that contains a large number of fractures. However, most of the previous numerical studies assumed identical hydraulic and mechanical apertures in DFN models for simplicity. In other words, a fracture network consisting of rough fractures is usually simplified as an idealized fracture network where individual fractures are smooth and parallel walled, without properly considering the reduced hydraulic apertures due to roughness (Fig. 1). Up to now, it still remains poorly understood that how the reduced hydraulic apertures in local fractures affect the macroscopic fluid flow and heat transfer in complex fracture systems.12–14 The main objective of this study is to examine the role of fracture surface roughness in coupled fluid flow and heat transfer in fractured rocks, because an accurate understanding of fluid flow and heat transfer through fracture networks in rocks is a critical
30
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 1. A rock fracture network (left, varying apertures in each single fracture) and an idealized discrete fracture network model (right, smooth and parallel walled fractures). Line thickness represents magnitudes of fracture apertures.
issue in many applications, such as underground nuclear waste repositories, CO2 sequestration, and enhanced geothermal systems. In a complex fracture network, surface roughness varies within different fractures, but there has not been a reported statistical distribution that can describe the roughness distribution in fracture networks. For this reason, we reviewed two published sets of fracture surface roughness data in order to derive a general statistical distribution of surface roughness in fracture networks. The influences of surface roughness of individual fractures on the macroscopic fluid flow and heat transfer in complex fracture networks were numerically studied.
0.14 Oskarshamn/Fotsmark data Normal fit
0.12
Density
0.1 0.08 0.06 0.04 0.02
2. Distribution of fracture surface roughness 0 0
2
4
6
8
10
12
14
16
18
20
JRC
0.16 Bakhtiary data Lognormal fit
0.14 0.12
Density
0.1 0.08 0.06 0.04 0.02 0 0
2
4
6
8
10
12
14
16
18
20
JRC Fig. 2. Observed and modeled distributions of fracture roughness coefficient (JRC) in rock mass. (a) Normal distribution for the Oskarshamn/Forsmark data.17 (b) Lognormal distribution for the Bakhtiary data.18 Table 1. Best-fit parameters of JRC distributions in rock mass. Parameters Site
JRC distribution
μ
σ
Oskarshamn/Forsmark, Sweden17 Bakhtiary dam, Iran18
Normal distribution Lognormal distribution
7.5 1.7
3.3 0.5
Since JRC was proposed together with the empirical criterion of joint shear strength by Barton,15 a number of studies have been devoted to accurately and objectively measure the values of JRC based on the ten standard profiles.9,16 However, the directly related question of JRC distribution in a field seems to have attracted very little attention, mainly due to the insufficient in-situ data. Two databases were reported in the recent studies: (1) Asadollahi 17 reviewed a series of site investigation reports published by Swedish Nuclear Fuel and Waste Management Co. (SKB, www.skb. com) that included direct shear tests on 175 rock fracture samples collected in Oskarshamn and Forsmark, Sweden, and back-calculated the JRC values based on Barton's failure criterion15; (2) Sanei et al.18 conducted direct shear tests on 106 rock fracture/bedding plane samples collected at the Bakhtiary dam site, Iran, as well as three in-situ direct shear tests on bedding planes at the same site, and estimated the JRC values based on the values of fractal dimension. With these two available databases, we attempted to answer the question whether a common statistical distribution that can describe the JRC/roughness distribution in fracture systems exists. The chi-square goodness-of-fit tests were applied to examine the appropriate statistical distributions of JRC/roughness in fracture networks. For the chi-square goodness-of-fit computation, the data are divided into k bins and the test statistic ( χ 2) is of the form k
χ2 =
∑ i=1
2
( Oi − Ei) Ei
(1)
where Oi and Ei are the observed and expected frequency for the ith bin, respectively. The expected frequency for the ith bin can be calculated by
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
31
Table 2. Summary of empirical models relating hydraulic and mechanical apertures.20 References
e/E
Patir and Cheng21
1/3 ⎡ ⎛ E ⎞⎤ ⎢ 1 − 0.9exp⎝⎜ −0.56 σ ⎠⎟⎥ ⎣ ⎦ E
Barton et al.2
⎛ ⎜ ⎝
Renshaw22
⎡ ⎢1 + ⎢⎣
Zimmerman and Bodvarsson4 Waite et al.23 Olsson and Barton24
1/2 e ⎞ ⎟ 2.5 JRC ⎠
−1/2 σ2⎤ E⎥ E2 ⎥
⎦
⎡ ⎤1/3 σ2 ⎢ 1 − 1.5 E + ⋅⋅⋅⎥ 2 ⎢⎣ ⎥⎦ E E is the harmonic mean of mechanical aperture; τ is tortuosity
E /τ1/3 ⎛ ⎜ ⎝
JRC 2 mob e
Rasouli and Hosseinian25 Xiong et al.26
Li and Jiang27
JRCmob is the mobilized Joint Roughness Coefficient; us is shear displacement; usp is peak
1/2 e ⎞ us ≤ 0.75usp ⎟ 2.5 JRC ⎠
us ≥ 0.75usp
⎡ −0.565 ⎢ 1 − 0.03dmc ⎣
(
)
⎡ ⎢ 1− ⎣
(
σE E
1 1 + Z 2.25 2
shear displacement
⎛
)⎜⎝ 1 −
σE E
1/3 JRCa ⎤
⎥ ⎦
σslope 10
dmc is minimum closure distance; JRCa is the average of joint roughness coefficients for σ 1/3 or ⎡⎣ 1 − 2.25 E ⎤⎦ upper and lower rock fracture profiles. E
Re < 1
1 1 + Z 2.25 + (0.00006 + 0.004Z 2.25)(Re − 1) 2 2
Ei = F ( x2) − F ( x1)
σslope is the standard deviation of local slope of fracture surface; Re is Reynolds number
1/3 ⎞⎤ Re ⎟⎥ ⎠⎦
Re ≥ 1
(2)
where F is the cumulative distribution function of the probability distribution being tested; x1 and x2 are the lower and upper limits. The observed JRC in the above two databases was fitted to some common probabilistic distributions, i.e., normal, lognormal, Weibull and Gamma distributions, using chi-square goodness-offit test. The best-fit distribution for the Oskarshamn/Forsmark data was normal distribution that yielded a chi-square test statistic of 3.82, whereas the best-fit distribution for the Bakhtiary data was lognormal distribution with a chi-square test statistic of 4.39. Note that a significance level of 10% was satisfied in both of these two fitting cases, which indicates that the normal and lognormal distributions provided a good representation of the observed data in Oskarshamn/Forsmark and Bakhtiary, respectively (Fig. 2). The best-fit parameters of the probability distributions are presented in Table 1. The mean value of JRC in Oskarshamn/Forsmark and Bakhtiary were 7.5 and 6.5, respectively, between which the difference was relatively small, but their probability distributions were obviously different (Fig. 2). This indicates that the JRC distribution is dependent on site geological condition, and it is difficult to conclude a uniform probability distribution that can be generally applicable to describe JRC distribution in rock masses. As a generic study, both normal and lognormal distributions were considered to investigate the role of fracture surface roughness in coupled fluid flow and heat transfer in fracture networks.
3. Modeling procedure 3.1. Relationship between mechanical and hydraulic apertures It is well known that the mechanical aperture of a rock fracture is usually larger than its hydraulic aperture, mainly due to fracture surface roughness.19 Zhao and Li20 reviewed the main empirical models between hydraulic aperture ( e ) and mechanical aperture ( E ), in terms of σE /E , JRC, Z2 and tortuosity ( τ ) (Table 2). Some of them include more parameters, in addition to the roughness parameter.25–27 Those modes were proposed based on different rock fracture samples, and thus they predict obviously different values of e/E for the same values of roughness parameters (Fig. 3). Generally, the models proposed by Patir and Cheng,21 Renshaw22 and Li and Jiang27 estimate relatively minor influence of surface roughness, i.e. larger values of e/E are given by these models. However, the models in Barton et al.2 and Olsson and Barton24 may give unrealistic values of e/E ( 41) for small JRC, and the models in Zimmerman and Bodvarsson4 and Rasouli and Hosseinian25 may give negative values of e/E for larger σE /E . The possible reasons mainly include: (1) Those empirical models were derived from a specific range of roughness, which cannot be extended to a general case; (2) The inertial effect of fluid becomes significant under the condition of high Reynolds numbers (Re), which, however, is not considered in the above models, except Xiong et al.26 and Li and Jiang27; (3) Matching state of fracture also affects the relationship between e and E , but this factor is not properly considered in the above models.
32
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
1.0
0.8
e/E
0.6
0.4
0.2
Patir and Cheng (1978) Renshaw (1995) Zimmerman and Bodvarsson (1996) Rasouli and Hosseinian (2011)
0.0 0.0
0.2
0.4
0.6
0.8
1.0
σ E /E Fig. 4. Three JRC distributions used for studying the effects of fracture surface roughness on the macroscopic fluid flow and heat transfer in fractured rocks. 1.0
⎛ κf ⎞ ∇T ⎜ − e∇T P ⎟ = 0 ⎝ η ⎠
0.8
(4)
e=1000μm
0.6
eCf
e /E
dc=100μm
∂T → + eρw c w u w ⋅∇T T − e∇T (λ f ∇T T ) = fupe + fbottome ∂t
(5) 2
where ∇T means tangential derivatives of field variables; κf = e /12 is the intrinsic permeability of fracture based on the Cubic law; e is the hydraulic aperture of a fracture; η is the water dynamic viscosity; P is hydraulic pressure; Cf is the volumetric heat capacity of the water in fractures; ρw is the water density; cw is the heat caκ → pacity of water; u w = ηf ∇T P is the darcy velocity of fluid in frac-
dc=50μm
0.4 Barton et al. (1985) Barton et al. (1985) Barton et al. (1985) Li and Jiang (2013) Rasouli and Hosseinian (2011) Rasouli and Hosseinian (2011) Rasouli and Hosseinian (2011)
0.2
e=100μm
e=10μm
0.0 0
5
tures; λf is the effective heat conduction coefficient of fluid; fupe
dc=10μm 10
15
20
JRC
Fig. 3. Comparison of empirical relationships between hydraulic aperture (e) and mechanical aperture (E) for rough rock fractures. (a) Roughness parameter – sE/E. (b) Roughness parameter – JRC or Z2.
3.2. Fluid flow and heat transfer simulations In this study, COMSOL Multiphysics was adopted to build up the DFN models and simulate fluid flow and heat transfer in crystalline fractured rocks. We considered single phase flow and heat transfer in fractures, and the permeability of rock matrix was neglected for simplicity. The energy conservation in rock matrix is governed by,
dT C + ∇( − λ eq∇T ) = 0 dt
(3)
where C is the volumetric heat capacity of the rock matrix; T is temperature; t is time; λeq is the effective heat conduction coefficient of the rock matrix. Local thermal equilibrium was assumed because of zero flow rates in the rock matrix. Mass conservation equation is not necessary for impermeable rock matrix. In the fracture networks, isolated fractures and dead-ends of fractures were removed. It was assumed that the distributions of hydraulic pressure and temperature along fracture's cross section were uniform. The governing equations of rock fractures are,
and fbottome are the energy exchange terms between the two fracture surfaces and rock matrix. Finite element method was used to solve (Eqs. (3)–(5)), and we could add sources and constraints on edges and points using weak form contributions. In particular, Eq. (5) was rewritten in weak form contributions to represent heat transfer between rock matrix and fracture. Rock matrix and fractures were discretized into finite triangle elements and line elements, respectively. When discretization of the domain and time-integration using implicit backward differentiation formulas were completed, the linear system was solved using parallel direct sparse solver interface in COMSOL based on LU decomposition. 3.3. Setup of discrete fracture network models Based on the statistical geometric parameters obtained from field mapping at the Sellafield area, Cumbria, UK, a square DFN model of side length 5 m was studied, which was above the Representative Elementary Volume (REV) size12 and was used to study the stress effects on macroscopic flow and transport in fracture networks.14 In this generic study, mechanical apertures were assumed to be a constant of 6.5, 65 and 650 mm for all fractures, respectively, and other geometrical properties can be found in Zhao et al.14 The fact that fracture surface roughness may vary among different fractures causes different hydraulic apertures in different fractures with the same mechanical aperture. The normal and lognormal distributions determined in Section 2 were used to describe the JRC distribution in the DFN model. Two empirical models relating hydraulic apertures to mechanical
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Case 1 + Barton
Case 1 + Li & Jiang
Case 2 + Barton
33
Case 2 + Li & Jiang
Case 4
Fig. 5. Hydraulic aperture distributions with varying JRC distributions and e/E models. Mechanical apertures are a constant of 65 mm for all fractures (Case 4), and fractures with hydraulic aperture less than 10 mm were not shown.
Table 3. Parameters for modeling fluid flow and heat transfer in fracture networks.
Fig. 6. Geometry of discrete fracture network model (side length of 5 m) and hydraulic and thermal boundary conditions.
apertures (Barton2; Li and Jiang)27 were considered. Note that the model in Li and Jiang27 is valid for JRC between 0 and 20, and the value of the hydraulic aperture of a fracture was forced to be smaller than the value of the mechanical aperture in Barton's model2 for unrealistic values of e/E (4 1). Li and Jiang's model27 is a function of Z2, and the values of Z2 can be obtained from JRC according to the relationship in Tse and Curden.7 In order to systematically study the macroscopic fluid flow and heat transfer in fractured rocks affected by fracture surface roughness, four cases were considered in this study (Fig. 4). (1) Case 1: Normal distribution of JRC with μ ¼7.5 and σ ¼3.3. (2) Case 2: Lognormal distribution of JRC with μ ¼1.7 and σ ¼0.5. (3) Case 3: Lognormal distribution of JRC with μ ¼2.0 and σ ¼0.24. (4) Case 4: Zero surface roughness (‘parallel plate’ model). The first two cases are designed based on the JRC distributions in Oskarshamn/Forsmark and Bakhtiary, respectively. Case 3 exhibited the mean and variance of JRC as Case 1, and thus normal and lognormal distributions of JRC could be compared excluding the influence of μ and σ . Case 4 without considering surface roughness served as a reference case. Five realizations of each case were utilized to avoid the influence of stochastic nature of the given JRC distributions. This study focused on the comparison between the same or different results considering different e/E models, and thus five realizations were sufficient to make the results statistically representative. Fu et al.28 also used five realizations to evaluate whether the results and conclusions were
Parameters
Value
Porosity of rock matrix Dynamic viscosity of fluid Density of fluid Heat capacity of fluid Heat conductivity of fluid Density of rock Heat capacity of rock Heat conductivity of rock
0.01 0.001 Pa s 1000 kg/m3 4200 J/(kg K) 0.5 W/(m K) 2700 kg/m3 800 J/(kg K) 3 W/(m K)
sensitive to random realizations. Compared with Li and Jiang's model,27 Barton's model2 generally gave smaller hydraulic apertures (Fig. 5). Fig. 6 shows the applied hydraulic boundary conditions: top and bottom boundaries were assumed to be impermeable, while the hydraulic pressures applied on the right and left boundaries generated a pressure gradient of 1 105 Pa/m in the horizontal direction. Heat transfer was modeled based on the calculated steady flow through fracture networks. We assumed that the initial temperature of rock matrix was 200 °C. Cold water of 50 °C was injected along the right boundary. The top and bottom boundaries were thermal insulation (Fig. 6). Parameters for modeling fluid flow and heat transfer in fracture networks are shown in Table 3.
4. Results 4.1. Macroscopic fluid flow This section first uses the results of fracture networks with mechanical apertures of 65 mm to illustrate the effect of surface roughness on macroscopic fluid flow in fracture networks, and then whether the mechanical aperture has an effect on the results is presented, by comparing the flow results of fracture networks with mechanical apertures of 6.5, 65 and 650 mm. A macroscopically horizontal flow occurred through the fracture network, and obvious channels with different magnitudes of flow rates were observed for all the cases (Fig. 7). The total flow rates along the outlet boundaries were normalized by the total flow rate of 4.76 10 5 m2/s of Case 4 and are compared in Fig. 8. Among Case 1–4, the greatest total flow rate occurred in Case 4, in which roughness was not considered. When using the relationship of e/E in Barton,2 the total flow rates reduced by 93.8%, 71.7% and 90.7% for Case 1, 2, and 3, respectively (columns b, c and d in Fig. 8). In addition, the total flow rate was more sensitive to the parameters
34
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 7. Flow rate distributions in DFN models with varying JRC distributions and e/E models (flow rates less than 10 7 m2/s were not shown).
Fig. 8. Comparison of the total flow rates at the outlet (left) boundary. The total flow rates for cases 1–3 were normalized by the flow rate (4.76 10 5 m2/s) of case 4. Note that the value of the total flow rate was an average of the five realizations for each case.
of JRC distribution when considering the relationship of e/E in Barton2 than that in Li and Jiang,27 i.e., a small deviation of the mean of JRC (1.4) between normal and lognormal JRC distributions induced significantly different flow rates (comparing the middle columns between Case 2 (column b) and 3 (column c) in Fig. 8). However, the difference between the total flow rates with normal and lognormal JRC distributions was relatively small if these two JRC distributions had an identical value of mean and variance (Comparing the middle columns between Case 1 (column a) and 3 (column c) in Fig. 8). Using the relationship of e/E in Li and Jiang,27 the total flow rates slightly decreased by 8.2%, 6.6% and 7.6% for Case 1, 2 and 3, respectively (columns e, f and g in Fig. 8). This
indicates that changes of the total flow rates were not significantly affected by the type or parameters of JRC distribution, if considering the e/E model of Li and Jiang27 in DFN modes. Fig. 9 illustrates the channeling of the five realizations of Case 1 (normal distribution of JRC) that employed Barton's e/E model.2 The five realizations exhibited obviously different spatial distributions of flow rate (the relatively high flow rate channels were marked by red ellipses), and they had different total flow rates at the outlet boundary. Another five realizations of Case 1 that considered Li and Jiang's e/E model27 had almost the same spatial distribution of flow rate and the variation in total flow rate is negligible. Table 4 also shows that the coefficients of variation of total flow rates for DFN models considering Barton's e/E relationship2 were much greater than those of DFN models considering the relationship of e/E in Li and Jiang.27 The effects of surface roughness on macroscopic fluid flow in fracture networks with mechanical apertures of 6.5, 65 and 650 mm are compared in Table 4. When considering Barton's e/E model,2 fracture surface roughness had a negligible influence on the flow rates in fracture networks with a mechanical aperture of 650 mm, whereas the total flow rates significantly reduced in fracture networks with a mechanical aperture of 6.5 mm. This can be explained by the fact shown in Fig. 3(b) that the effect of surface roughness became less significant for greater mechanical apertures in Barton's model 2. In contrast, considering Li and Jiang's e/E model27 caused that the effects of fracture surface roughness on fluid flow in fracture networks were always small and independent on mechanical apertures. This is due to the fact that Li and Jiang's e/E model27 was independent on E (Fig. 3(b)). 4.2. Macroscopic heat transfer This section uses the results of fracture networks with mechanical apertures of 65 mm to illustrate the effect of surface
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
35
2.81× 10-6 m2/s
2.97× 10-6 m2/s
2.95× 10-6 m2/s
3.52× 10-6 m2/s
2.50× 10-6 m2/s
4.37× 10-5 m2/s
4.37× 10-5 m2/s
4.36× 10-5 m2/s
4.38× 10-5 m2/s
4.37× 10-5 m2/s
Fig. 9. Flow rates distributions of 5 realizations considering normal distribution of JRC (Case 1). The channels with high flow rates are marked by red ellipses, and the total flow rates are indicated below the images. (a) Considering the e/E model in Barton.2 (b) Considering the e/E model in Li and Jiang.27 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4. Mean and coefficient of variation of total flow rates for different DFN models. Mean mechanical Category of model aperture (μm)
Coefficient of Normalized mean of flow rates (m2/s) variation for flow rates (%)
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
6.9 10 5 6.2 10 4 9.6 10 5 0.92 0.93 0.92
13.0 17.2 9.2 0.2 0.2 0.2
65
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
0.06 0.28 0.09 0.92 0.93 0.92
11.2 9.3 8.9 0.2 0.2 0.2
650
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
0.98 0.96 1.00 0.92 0.93 0.92
0.4 1.3 0.4 0.2 0.2 0.2
6.5
roughness on macroscopic heat transfer in fracture networks. The temperature breakthrough curves (outlet boundaries at the left sides of the DFN models) of the four cases are shown in Fig. 10. Due to the smaller flow rates in the DFN models considering Barton's e/E model,2 the breakthrough curves were less steep and
Fig. 10. Temperature break through curve (outflow borders at the left side of DFN models).
exhibited a long tail effect compared with the DFN models considering Li and Jiang's e/E model.27 From the temperature distributions in DFN models of different e/E model (Fig. 11), it was observed that thermal front moved more slowly in DFN models considering Barton's e/E model.2 Temperature distributions in each realization of Case 1 are compared in Fig. 12, and the comparison focuses on the same or different temperature distributions considering Barton's e/E
36
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 11. Temperature distributions in DFN models with varying JRC distributions and e/E models (t ¼1 105 s).
Fig. 12. Temperature distributions in 5 realizations considering normal distribution of JRC (Case 1). (a) Considering the e/E model in Barton2 (t¼ 5 105 s). (b) Considering the e/E model in Li and Jiang27 (t ¼ 1 105 s).
model2 and Li and Jiang's e/E model,27 respectively. Due to the channeling (heterogeneity of flow rates, Fig. 9), the temperature distributions in each realization of Case 1 considering Barton's e/E model2 were slightly different. In contrast, the temperature distributions in each realization of Case 1 considering Li and Jiang's
e/E model27 were almost the same. This phenomenon was also reflected in the temperature breakthrough curves for different realizations of Case 1 (Fig. 13). Therefore, heat transfer in fracture networks is dependent on flow rates that are affected by the e/E models.
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
37
transfer in DFN models considering Li and Jiang's e/E model27 would be neglected for a first order approximation. However, Barton's e/E model 2 usually estimates small values of e/E ratio, so fracture surface roughness would play an important role in fluid flow and heat transfer in DFN models considering Barton's e/E model.2 In addition, Barton's e/E model2 is dependent on the mechanical apertures. Therefore, a proper e/E model should be chosen to accurately model the coupled fluid flow and heat transfer processes in fractured rocks. (3) Macroscopic fluid flow and heat transfer in different realizations of DFN models considering Barton's e/E model2 are different from each other, and thus a number of realizations are needed to reflect the fluid flow and heat transfer behavior in this case. If considering Li and Jiang's e/E model,27 one realization is sufficient to predict the fluid flow and heat transfer processes in DFN models. Lang et al.29 showed that two dimensional analysis cannot be used to approximately estimate the three dimensional equivalent permeability of fractured rock mass. Therefore, the present conclusions based on two dimensional simulations may or may not be extended to three dimensional case, and a systematic verification is needed. In this study, we considered the macroscopic fluid flow and heat transfer affected by fracture surface roughness. In practice, fracture roughness may vary due to other mechanical processes, and gouge materials may fill the fracture voids. Guo et al. 30 pointed out that thermal stress can also play an important significant role in flow pattern evolution and heat production. These factors can influence the fracture permeability and consequently the heat transfer and are important in the coupled THMC modeling, but they are beyond the scope of this paper. The relationships used in this study are mainly derived based on the equivalent fluid flow calculation, without considering the effects of tortuosity. This approximation may be valid for modeling of macroscopic fluid flow, but it may underestimate the thermal transfer area in fractures. To better understand in-situ problems, further studies on mechanical-hydraulic aperture relationship and roughness distribution are strongly recommended in the future.
Acknowledgments
Fig. 13. Temperature breakthrough curves for the 5 realizations considering normal distributions of JRC (Case 1). (a) Considering the e/E model in Barton.2 (b) Considering the e/E model in Li and Jiang.27
5. Conclusion and implication Based on the empirical relationships of mechanical-hydraulic apertures and the estimated distributions of fracture surface roughness (JRC), the role of fracture roughness distribution in macroscopic fracture flow and heat transfer through fractured rocks was numerically investigated. The results show that both flow rates and heat breakthrough curves can be affected to some extent by different mechanical-hydraulic aperture relationships and various roughness distributions. We can come to the main conclusions as follows: (1) Based on the JRC data in Oskarshamn/Forsmark and Bakhtiary, normal and lognormal distributions can be used to properly describe the JRC distributions in these two sites, respectively. DFN models with the normal distribution of JRC are less permeable than those with the lognormal distribution of JRC. As a result, the movement of thermal front is relatively slower in DFN models with the normal distribution of JRC. (2) Li and Jiang's e/E model27 predicts a relatively small influence of surface roughness (large values of e/E ratio), and thus the effects of fracture surface roughness on fluid flow and heat
This work was jointly supported by the National Natural Science Foundation of China (Nos. 41272279, 51322401 and 51509138), Beijing Natural Science Foundation (No. 8152020), the Recruitment Program of Beijing Youth Experts (2014000020124G115), the Openend Research Fund of the State Key Laboratory for Geomechanics and Deep Underground Engineering (No. SKLGDUEK1407), and the Scientific Research Foundation of the Education Ministry for Returned Chinese Scholars, China.
References 1. Grasselli G, Wirth J, Egger P. Quantitative three-dimensional description of a rough surface and parameter evolution with shearing. Int J Rock Mech Min Sci. 2002;39:789–800. 2. Barton N, Bandis S, Bakhtar K. Strength, deformation and conductivity coupling of rock joints. Int J Rock Mech Min Sci. 1985;22:121–140. 3. Myers N. Characterization of surface roughness. Wear. 1962;5:182–189. 4. Zimmerman RW, Bodvarsson GS. Hydraulic conductivity of rock fractures. Transp Porous Med. 1996;23:1–30. 5. Huang SL, Oelfke SM, Speck RC. Applicability of fractal characterization and modelling to rock joint profiles. Int J Rock Mech Min Sci. 1992;29:135–153. 6. Tatone BSA, Grasselli G. A new 2D discontinuity roughness parameter and its correlation with JRC. Int J Rock Mech Min Sci. 2010;47:1391–1400. 7. Tse R, Cruden DM. Estimating joint roughness coefficients. Int J Rock Mech Min Sci Geomech Abstr. 1979;16:303–307. 8. Yu X, Vayssade B. Joint profiles and their roughness parameters. Int J Rock Mech Min Sci Geomech Abstr. 1991;28:333–336.
38
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
9. Gao Y, Wong LNY. A modified correlation between roughness parameter Z2 and the JRC. Rock Mech Rock Eng. 2015;48:387–396. 10. Li Y, Huang R. Relationship between joint roughness coefficient and fractal dimension of rock fracture surfaces. Int J Rock Mech Min Sci. 2015;75:15–22. 11. Boutt DF, Grasselli G, Fredrich JT, Cook BK, Williams JR. Trapping zones: the effect of fracture roughness on the directional anisotropy of fluid flow and colloid transport in a single fracture. Geophys Res Lett. 2006;33:L21402. 12. Min KB, Jing L, Stephansson O. Fracture system characterization and evaluation of the equivalent permeability tensor of fractured rock masses using a stochastic REV approach. Hydrogeol J. 2004;12:497–510. 13. Shaik AR, Rahman SS, Tran NH, Tran T. Numerical simulation of fluid-rock coupling heat transfer in naturally fractured geothermal system. Appl Therm Eng. 2011;31:1600–1606. 14. Zhao Z, Jing L, Neretnieks I, Moreno L. Numerical modeling of stress effects on solute transport in fractured rocks. Comput Geotech. 2011;38:113–126. 15. Barton N. Review of a new shear-strength criterion for rock joints. Eng Geol. 1973;7:287–332. 16. Barton N, Choubey V. The shear strength of rock joints in theory and practice. Rock Mech. 1977;10:1–54. 17. Asadollahi P. Stability Analysis of a Single Three Dimensional Rock Block: Effect of Dilatancy and High-Velocity Water Jet Impact [PhD Thesis]. Austin: UTAU; 2009. 18. Sanei M, Faramarzi L, Fahimifar A, Goli S, Mehinrad A, Rahmati A. Shear strength of discontinuities in sedimentary rock masses based on direct shear tests. Int J Rock Mech Min Sci. 2015;75:119–131. 19. Tsang YW, Witherspoon PA. The dependence of fracture mechanical and fluid flow properties on fracture roughness and sample size. J Geophys Res. 1983;88:2359–2366. 20. Zhao Z, Li B. On the role of fracture surface roughness in fluid flow and solute transport through fractured rocks. In: Proceeding of the 13th International
Congress of Rock Mechanics. Montreal, Canada; 10–13 May 2015. 21. Patir N, Cheng HS. An average flow model for determining effects of threedimensional roughness on partial hydrodynamic lubrication. J Lubric Tech. 1978;100:12–17. 22. Renshaw CE. On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J Geophys Res. 1995;100:24629–24636. 23. Waite M, Ge S, Spetzler H. A new conceptual model for flow in discrete fractures: An experimental and numerical study. J Geophys Res. 1999;104:13049– 13059. 24. Olsson R, Barton N. An improved model for hydromechanical coupling during shearing of rock joints. Int J Rock Mech Min Sci. 2001;38:317–329. 25. Rasouli V, Hosseinian A. Correlations developed for estimation of hydraulic parameters of rough fractures through the simulation of JRC flow channels. Rock Mech Rock Eng. 2011;44:447–461. 26. Xiong X, Li B, Jiang Y, Koyama T, Zhang C. Experimental and numerical study of the geometrical and hydraulic characteristics of a single rock fracture during shear. Int J Rock Mech Min Sci. 2011;48:1292–1302. 27. Li B, Jiang Y. Quantitative estimation of fluid flow mechanism in rock fracture taking into account the influences of JRC and Reynolds number. J MMIJ. 2013;129:479–484 [in Japanese]. 28. Fu P, Hao Y, Walsh SDC, Carrigan CR. Thermal drawdown-induced flow channeling in fracture geothermal Reservoirs. Rock Mech Rock Eng. 2016;49:1001– 1024. 29. Guo B, Fu P, Hao Y, Peters CA, Carrigan CR. Thermal drawdown-induced flow channeling in a single fracture in EGS. Geothermics. 2016;61:46–62. 30. Lang P, Paluszny A, Zimmerman R. Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J Geophys Res. 2014;119:6288–6307.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
The role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks Shuang Luo a, Zhihong Zhao a,n, Huan Peng a, Hai Pu b a b
Department of Civil Engineering, Tsinghua University, Beijing 100084, China State Key Laboratory for Geomchanics & Deep Underground Engineering, China University of Mining & Technology, Xuzhou, Jiangsu 221116, China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 12 November 2015 Received in revised form 13 April 2016 Accepted 9 May 2016
Rock fractures are major conduits for fluid flow in fractured rocks, and the convective heat transfer between rock fracture surfaces and circulating fluid is a critical issue in heat recovery in fractured rocks. It has been demonstrated that fracture surface roughness has a significant influence on the mechanical, hydraulic, thermal and transport behavior of single fractures. This study aimed to assess the effects of local surface roughness of fractures on fluid flow and heat transfer processes at the macroscopic scale of fracture networks. Two distributions of Joint Roughness Coefficient (JRC) were determined based on the JRC data in Oskarshamn/Forsmark, Sweden and Bakhtiary, Iran. Two empirical models relating hydraulic apertures to mechanical apertures were considered. A total of ninety-one realizations that considered different JRC distributions and empirical models of mechanical-hydraulic apertures were studied. The results show that fracture surface roughness can affect the fluid flow and heat transfer processes in fracture networks to various extents, mainly depending on the empirical models of mechanical-hydraulic apertures. In other words, the role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks is critical, when using a model of mechanical-hydraulic apertures that predicts significant reduced hydraulic apertures. Discrete fracture networks models with the normal distribution of JRC are less permeable than those with the lognormal distribution of JRC, using the fitting parameters of in-situ JRC data. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Rock fracture Surface roughness Fluid flow Heat transfer Discrete fracture network model
1. Introduction A rock fracture consists of two rough surfaces that are in contact with each other at some locations, but separated at others. Roughness refers to the local departures from planarity,1 and a number of parameters have been proposed to characterize the fracture surface roughness, e.g., Joint Roughness Coefficient (JRC),2 Z2,3 the ratio of standard deviation ( σE ) of the varying aperture over mechanical aperture ( E ) ( σE /E ),4 and fractal dimension.5 Grasselli and his co-workers proposed a mathema* /(C + 1) to parameterize fracture surface tical expression of θmax roughness, which can be used for three-dimensional surface topography1 or two-dimensional profiles.6 The regressive relationships between these roughness parameters have also been extensively investigated .6–10 Fractures in rock masses have a controlling influence on the mechanical behavior of rock masses, since they provide planes of weakness on which further displacement can more readily occur. n
Corresponding author. E-mail address: [email protected] (Z. Zhao).
http://dx.doi.org/10.1016/j.ijrmms.2016.05.006 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
Fractures also often provide major conduits through which groundwater can flow. It has been demonstrated that fracture surface roughness has a significant influence on the mechanical, hydraulic, thermal and transport behavior of single fractures.11 Discrete fracture network (DFN) models have been widely used to simulate the coupled thermal-hydrological-mechanical-chemical (THMC) processes on the field scale of fractured rock mass that contains a large number of fractures. However, most of the previous numerical studies assumed identical hydraulic and mechanical apertures in DFN models for simplicity. In other words, a fracture network consisting of rough fractures is usually simplified as an idealized fracture network where individual fractures are smooth and parallel walled, without properly considering the reduced hydraulic apertures due to roughness (Fig. 1). Up to now, it still remains poorly understood that how the reduced hydraulic apertures in local fractures affect the macroscopic fluid flow and heat transfer in complex fracture systems.12–14 The main objective of this study is to examine the role of fracture surface roughness in coupled fluid flow and heat transfer in fractured rocks, because an accurate understanding of fluid flow and heat transfer through fracture networks in rocks is a critical
30
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 1. A rock fracture network (left, varying apertures in each single fracture) and an idealized discrete fracture network model (right, smooth and parallel walled fractures). Line thickness represents magnitudes of fracture apertures.
issue in many applications, such as underground nuclear waste repositories, CO2 sequestration, and enhanced geothermal systems. In a complex fracture network, surface roughness varies within different fractures, but there has not been a reported statistical distribution that can describe the roughness distribution in fracture networks. For this reason, we reviewed two published sets of fracture surface roughness data in order to derive a general statistical distribution of surface roughness in fracture networks. The influences of surface roughness of individual fractures on the macroscopic fluid flow and heat transfer in complex fracture networks were numerically studied.
0.14 Oskarshamn/Fotsmark data Normal fit
0.12
Density
0.1 0.08 0.06 0.04 0.02
2. Distribution of fracture surface roughness 0 0
2
4
6
8
10
12
14
16
18
20
JRC
0.16 Bakhtiary data Lognormal fit
0.14 0.12
Density
0.1 0.08 0.06 0.04 0.02 0 0
2
4
6
8
10
12
14
16
18
20
JRC Fig. 2. Observed and modeled distributions of fracture roughness coefficient (JRC) in rock mass. (a) Normal distribution for the Oskarshamn/Forsmark data.17 (b) Lognormal distribution for the Bakhtiary data.18 Table 1. Best-fit parameters of JRC distributions in rock mass. Parameters Site
JRC distribution
μ
σ
Oskarshamn/Forsmark, Sweden17 Bakhtiary dam, Iran18
Normal distribution Lognormal distribution
7.5 1.7
3.3 0.5
Since JRC was proposed together with the empirical criterion of joint shear strength by Barton,15 a number of studies have been devoted to accurately and objectively measure the values of JRC based on the ten standard profiles.9,16 However, the directly related question of JRC distribution in a field seems to have attracted very little attention, mainly due to the insufficient in-situ data. Two databases were reported in the recent studies: (1) Asadollahi 17 reviewed a series of site investigation reports published by Swedish Nuclear Fuel and Waste Management Co. (SKB, www.skb. com) that included direct shear tests on 175 rock fracture samples collected in Oskarshamn and Forsmark, Sweden, and back-calculated the JRC values based on Barton's failure criterion15; (2) Sanei et al.18 conducted direct shear tests on 106 rock fracture/bedding plane samples collected at the Bakhtiary dam site, Iran, as well as three in-situ direct shear tests on bedding planes at the same site, and estimated the JRC values based on the values of fractal dimension. With these two available databases, we attempted to answer the question whether a common statistical distribution that can describe the JRC/roughness distribution in fracture systems exists. The chi-square goodness-of-fit tests were applied to examine the appropriate statistical distributions of JRC/roughness in fracture networks. For the chi-square goodness-of-fit computation, the data are divided into k bins and the test statistic ( χ 2) is of the form k
χ2 =
∑ i=1
2
( Oi − Ei) Ei
(1)
where Oi and Ei are the observed and expected frequency for the ith bin, respectively. The expected frequency for the ith bin can be calculated by
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
31
Table 2. Summary of empirical models relating hydraulic and mechanical apertures.20 References
e/E
Patir and Cheng21
1/3 ⎡ ⎛ E ⎞⎤ ⎢ 1 − 0.9exp⎝⎜ −0.56 σ ⎠⎟⎥ ⎣ ⎦ E
Barton et al.2
⎛ ⎜ ⎝
Renshaw22
⎡ ⎢1 + ⎢⎣
Zimmerman and Bodvarsson4 Waite et al.23 Olsson and Barton24
1/2 e ⎞ ⎟ 2.5 JRC ⎠
−1/2 σ2⎤ E⎥ E2 ⎥
⎦
⎡ ⎤1/3 σ2 ⎢ 1 − 1.5 E + ⋅⋅⋅⎥ 2 ⎢⎣ ⎥⎦ E E is the harmonic mean of mechanical aperture; τ is tortuosity
E /τ1/3 ⎛ ⎜ ⎝
JRC 2 mob e
Rasouli and Hosseinian25 Xiong et al.26
Li and Jiang27
JRCmob is the mobilized Joint Roughness Coefficient; us is shear displacement; usp is peak
1/2 e ⎞ us ≤ 0.75usp ⎟ 2.5 JRC ⎠
us ≥ 0.75usp
⎡ −0.565 ⎢ 1 − 0.03dmc ⎣
(
)
⎡ ⎢ 1− ⎣
(
σE E
1 1 + Z 2.25 2
shear displacement
⎛
)⎜⎝ 1 −
σE E
1/3 JRCa ⎤
⎥ ⎦
σslope 10
dmc is minimum closure distance; JRCa is the average of joint roughness coefficients for σ 1/3 or ⎡⎣ 1 − 2.25 E ⎤⎦ upper and lower rock fracture profiles. E
Re < 1
1 1 + Z 2.25 + (0.00006 + 0.004Z 2.25)(Re − 1) 2 2
Ei = F ( x2) − F ( x1)
σslope is the standard deviation of local slope of fracture surface; Re is Reynolds number
1/3 ⎞⎤ Re ⎟⎥ ⎠⎦
Re ≥ 1
(2)
where F is the cumulative distribution function of the probability distribution being tested; x1 and x2 are the lower and upper limits. The observed JRC in the above two databases was fitted to some common probabilistic distributions, i.e., normal, lognormal, Weibull and Gamma distributions, using chi-square goodness-offit test. The best-fit distribution for the Oskarshamn/Forsmark data was normal distribution that yielded a chi-square test statistic of 3.82, whereas the best-fit distribution for the Bakhtiary data was lognormal distribution with a chi-square test statistic of 4.39. Note that a significance level of 10% was satisfied in both of these two fitting cases, which indicates that the normal and lognormal distributions provided a good representation of the observed data in Oskarshamn/Forsmark and Bakhtiary, respectively (Fig. 2). The best-fit parameters of the probability distributions are presented in Table 1. The mean value of JRC in Oskarshamn/Forsmark and Bakhtiary were 7.5 and 6.5, respectively, between which the difference was relatively small, but their probability distributions were obviously different (Fig. 2). This indicates that the JRC distribution is dependent on site geological condition, and it is difficult to conclude a uniform probability distribution that can be generally applicable to describe JRC distribution in rock masses. As a generic study, both normal and lognormal distributions were considered to investigate the role of fracture surface roughness in coupled fluid flow and heat transfer in fracture networks.
3. Modeling procedure 3.1. Relationship between mechanical and hydraulic apertures It is well known that the mechanical aperture of a rock fracture is usually larger than its hydraulic aperture, mainly due to fracture surface roughness.19 Zhao and Li20 reviewed the main empirical models between hydraulic aperture ( e ) and mechanical aperture ( E ), in terms of σE /E , JRC, Z2 and tortuosity ( τ ) (Table 2). Some of them include more parameters, in addition to the roughness parameter.25–27 Those modes were proposed based on different rock fracture samples, and thus they predict obviously different values of e/E for the same values of roughness parameters (Fig. 3). Generally, the models proposed by Patir and Cheng,21 Renshaw22 and Li and Jiang27 estimate relatively minor influence of surface roughness, i.e. larger values of e/E are given by these models. However, the models in Barton et al.2 and Olsson and Barton24 may give unrealistic values of e/E ( 41) for small JRC, and the models in Zimmerman and Bodvarsson4 and Rasouli and Hosseinian25 may give negative values of e/E for larger σE /E . The possible reasons mainly include: (1) Those empirical models were derived from a specific range of roughness, which cannot be extended to a general case; (2) The inertial effect of fluid becomes significant under the condition of high Reynolds numbers (Re), which, however, is not considered in the above models, except Xiong et al.26 and Li and Jiang27; (3) Matching state of fracture also affects the relationship between e and E , but this factor is not properly considered in the above models.
32
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
1.0
0.8
e/E
0.6
0.4
0.2
Patir and Cheng (1978) Renshaw (1995) Zimmerman and Bodvarsson (1996) Rasouli and Hosseinian (2011)
0.0 0.0
0.2
0.4
0.6
0.8
1.0
σ E /E Fig. 4. Three JRC distributions used for studying the effects of fracture surface roughness on the macroscopic fluid flow and heat transfer in fractured rocks. 1.0
⎛ κf ⎞ ∇T ⎜ − e∇T P ⎟ = 0 ⎝ η ⎠
0.8
(4)
e=1000μm
0.6
eCf
e /E
dc=100μm
∂T → + eρw c w u w ⋅∇T T − e∇T (λ f ∇T T ) = fupe + fbottome ∂t
(5) 2
where ∇T means tangential derivatives of field variables; κf = e /12 is the intrinsic permeability of fracture based on the Cubic law; e is the hydraulic aperture of a fracture; η is the water dynamic viscosity; P is hydraulic pressure; Cf is the volumetric heat capacity of the water in fractures; ρw is the water density; cw is the heat caκ → pacity of water; u w = ηf ∇T P is the darcy velocity of fluid in frac-
dc=50μm
0.4 Barton et al. (1985) Barton et al. (1985) Barton et al. (1985) Li and Jiang (2013) Rasouli and Hosseinian (2011) Rasouli and Hosseinian (2011) Rasouli and Hosseinian (2011)
0.2
e=100μm
e=10μm
0.0 0
5
tures; λf is the effective heat conduction coefficient of fluid; fupe
dc=10μm 10
15
20
JRC
Fig. 3. Comparison of empirical relationships between hydraulic aperture (e) and mechanical aperture (E) for rough rock fractures. (a) Roughness parameter – sE/E. (b) Roughness parameter – JRC or Z2.
3.2. Fluid flow and heat transfer simulations In this study, COMSOL Multiphysics was adopted to build up the DFN models and simulate fluid flow and heat transfer in crystalline fractured rocks. We considered single phase flow and heat transfer in fractures, and the permeability of rock matrix was neglected for simplicity. The energy conservation in rock matrix is governed by,
dT C + ∇( − λ eq∇T ) = 0 dt
(3)
where C is the volumetric heat capacity of the rock matrix; T is temperature; t is time; λeq is the effective heat conduction coefficient of the rock matrix. Local thermal equilibrium was assumed because of zero flow rates in the rock matrix. Mass conservation equation is not necessary for impermeable rock matrix. In the fracture networks, isolated fractures and dead-ends of fractures were removed. It was assumed that the distributions of hydraulic pressure and temperature along fracture's cross section were uniform. The governing equations of rock fractures are,
and fbottome are the energy exchange terms between the two fracture surfaces and rock matrix. Finite element method was used to solve (Eqs. (3)–(5)), and we could add sources and constraints on edges and points using weak form contributions. In particular, Eq. (5) was rewritten in weak form contributions to represent heat transfer between rock matrix and fracture. Rock matrix and fractures were discretized into finite triangle elements and line elements, respectively. When discretization of the domain and time-integration using implicit backward differentiation formulas were completed, the linear system was solved using parallel direct sparse solver interface in COMSOL based on LU decomposition. 3.3. Setup of discrete fracture network models Based on the statistical geometric parameters obtained from field mapping at the Sellafield area, Cumbria, UK, a square DFN model of side length 5 m was studied, which was above the Representative Elementary Volume (REV) size12 and was used to study the stress effects on macroscopic flow and transport in fracture networks.14 In this generic study, mechanical apertures were assumed to be a constant of 6.5, 65 and 650 mm for all fractures, respectively, and other geometrical properties can be found in Zhao et al.14 The fact that fracture surface roughness may vary among different fractures causes different hydraulic apertures in different fractures with the same mechanical aperture. The normal and lognormal distributions determined in Section 2 were used to describe the JRC distribution in the DFN model. Two empirical models relating hydraulic apertures to mechanical
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Case 1 + Barton
Case 1 + Li & Jiang
Case 2 + Barton
33
Case 2 + Li & Jiang
Case 4
Fig. 5. Hydraulic aperture distributions with varying JRC distributions and e/E models. Mechanical apertures are a constant of 65 mm for all fractures (Case 4), and fractures with hydraulic aperture less than 10 mm were not shown.
Table 3. Parameters for modeling fluid flow and heat transfer in fracture networks.
Fig. 6. Geometry of discrete fracture network model (side length of 5 m) and hydraulic and thermal boundary conditions.
apertures (Barton2; Li and Jiang)27 were considered. Note that the model in Li and Jiang27 is valid for JRC between 0 and 20, and the value of the hydraulic aperture of a fracture was forced to be smaller than the value of the mechanical aperture in Barton's model2 for unrealistic values of e/E (4 1). Li and Jiang's model27 is a function of Z2, and the values of Z2 can be obtained from JRC according to the relationship in Tse and Curden.7 In order to systematically study the macroscopic fluid flow and heat transfer in fractured rocks affected by fracture surface roughness, four cases were considered in this study (Fig. 4). (1) Case 1: Normal distribution of JRC with μ ¼7.5 and σ ¼3.3. (2) Case 2: Lognormal distribution of JRC with μ ¼1.7 and σ ¼0.5. (3) Case 3: Lognormal distribution of JRC with μ ¼2.0 and σ ¼0.24. (4) Case 4: Zero surface roughness (‘parallel plate’ model). The first two cases are designed based on the JRC distributions in Oskarshamn/Forsmark and Bakhtiary, respectively. Case 3 exhibited the mean and variance of JRC as Case 1, and thus normal and lognormal distributions of JRC could be compared excluding the influence of μ and σ . Case 4 without considering surface roughness served as a reference case. Five realizations of each case were utilized to avoid the influence of stochastic nature of the given JRC distributions. This study focused on the comparison between the same or different results considering different e/E models, and thus five realizations were sufficient to make the results statistically representative. Fu et al.28 also used five realizations to evaluate whether the results and conclusions were
Parameters
Value
Porosity of rock matrix Dynamic viscosity of fluid Density of fluid Heat capacity of fluid Heat conductivity of fluid Density of rock Heat capacity of rock Heat conductivity of rock
0.01 0.001 Pa s 1000 kg/m3 4200 J/(kg K) 0.5 W/(m K) 2700 kg/m3 800 J/(kg K) 3 W/(m K)
sensitive to random realizations. Compared with Li and Jiang's model,27 Barton's model2 generally gave smaller hydraulic apertures (Fig. 5). Fig. 6 shows the applied hydraulic boundary conditions: top and bottom boundaries were assumed to be impermeable, while the hydraulic pressures applied on the right and left boundaries generated a pressure gradient of 1 105 Pa/m in the horizontal direction. Heat transfer was modeled based on the calculated steady flow through fracture networks. We assumed that the initial temperature of rock matrix was 200 °C. Cold water of 50 °C was injected along the right boundary. The top and bottom boundaries were thermal insulation (Fig. 6). Parameters for modeling fluid flow and heat transfer in fracture networks are shown in Table 3.
4. Results 4.1. Macroscopic fluid flow This section first uses the results of fracture networks with mechanical apertures of 65 mm to illustrate the effect of surface roughness on macroscopic fluid flow in fracture networks, and then whether the mechanical aperture has an effect on the results is presented, by comparing the flow results of fracture networks with mechanical apertures of 6.5, 65 and 650 mm. A macroscopically horizontal flow occurred through the fracture network, and obvious channels with different magnitudes of flow rates were observed for all the cases (Fig. 7). The total flow rates along the outlet boundaries were normalized by the total flow rate of 4.76 10 5 m2/s of Case 4 and are compared in Fig. 8. Among Case 1–4, the greatest total flow rate occurred in Case 4, in which roughness was not considered. When using the relationship of e/E in Barton,2 the total flow rates reduced by 93.8%, 71.7% and 90.7% for Case 1, 2, and 3, respectively (columns b, c and d in Fig. 8). In addition, the total flow rate was more sensitive to the parameters
34
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 7. Flow rate distributions in DFN models with varying JRC distributions and e/E models (flow rates less than 10 7 m2/s were not shown).
Fig. 8. Comparison of the total flow rates at the outlet (left) boundary. The total flow rates for cases 1–3 were normalized by the flow rate (4.76 10 5 m2/s) of case 4. Note that the value of the total flow rate was an average of the five realizations for each case.
of JRC distribution when considering the relationship of e/E in Barton2 than that in Li and Jiang,27 i.e., a small deviation of the mean of JRC (1.4) between normal and lognormal JRC distributions induced significantly different flow rates (comparing the middle columns between Case 2 (column b) and 3 (column c) in Fig. 8). However, the difference between the total flow rates with normal and lognormal JRC distributions was relatively small if these two JRC distributions had an identical value of mean and variance (Comparing the middle columns between Case 1 (column a) and 3 (column c) in Fig. 8). Using the relationship of e/E in Li and Jiang,27 the total flow rates slightly decreased by 8.2%, 6.6% and 7.6% for Case 1, 2 and 3, respectively (columns e, f and g in Fig. 8). This
indicates that changes of the total flow rates were not significantly affected by the type or parameters of JRC distribution, if considering the e/E model of Li and Jiang27 in DFN modes. Fig. 9 illustrates the channeling of the five realizations of Case 1 (normal distribution of JRC) that employed Barton's e/E model.2 The five realizations exhibited obviously different spatial distributions of flow rate (the relatively high flow rate channels were marked by red ellipses), and they had different total flow rates at the outlet boundary. Another five realizations of Case 1 that considered Li and Jiang's e/E model27 had almost the same spatial distribution of flow rate and the variation in total flow rate is negligible. Table 4 also shows that the coefficients of variation of total flow rates for DFN models considering Barton's e/E relationship2 were much greater than those of DFN models considering the relationship of e/E in Li and Jiang.27 The effects of surface roughness on macroscopic fluid flow in fracture networks with mechanical apertures of 6.5, 65 and 650 mm are compared in Table 4. When considering Barton's e/E model,2 fracture surface roughness had a negligible influence on the flow rates in fracture networks with a mechanical aperture of 650 mm, whereas the total flow rates significantly reduced in fracture networks with a mechanical aperture of 6.5 mm. This can be explained by the fact shown in Fig. 3(b) that the effect of surface roughness became less significant for greater mechanical apertures in Barton's model 2. In contrast, considering Li and Jiang's e/E model27 caused that the effects of fracture surface roughness on fluid flow in fracture networks were always small and independent on mechanical apertures. This is due to the fact that Li and Jiang's e/E model27 was independent on E (Fig. 3(b)). 4.2. Macroscopic heat transfer This section uses the results of fracture networks with mechanical apertures of 65 mm to illustrate the effect of surface
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
35
2.81× 10-6 m2/s
2.97× 10-6 m2/s
2.95× 10-6 m2/s
3.52× 10-6 m2/s
2.50× 10-6 m2/s
4.37× 10-5 m2/s
4.37× 10-5 m2/s
4.36× 10-5 m2/s
4.38× 10-5 m2/s
4.37× 10-5 m2/s
Fig. 9. Flow rates distributions of 5 realizations considering normal distribution of JRC (Case 1). The channels with high flow rates are marked by red ellipses, and the total flow rates are indicated below the images. (a) Considering the e/E model in Barton.2 (b) Considering the e/E model in Li and Jiang.27 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4. Mean and coefficient of variation of total flow rates for different DFN models. Mean mechanical Category of model aperture (μm)
Coefficient of Normalized mean of flow rates (m2/s) variation for flow rates (%)
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
6.9 10 5 6.2 10 4 9.6 10 5 0.92 0.93 0.92
13.0 17.2 9.2 0.2 0.2 0.2
65
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
0.06 0.28 0.09 0.92 0.93 0.92
11.2 9.3 8.9 0.2 0.2 0.2
650
Case Case Case Case Case Case
1-Barton 2-Barton 3-Barton 1-Li and Jiang 2-Li and Jiang 3-Li and Jiang
0.98 0.96 1.00 0.92 0.93 0.92
0.4 1.3 0.4 0.2 0.2 0.2
6.5
roughness on macroscopic heat transfer in fracture networks. The temperature breakthrough curves (outlet boundaries at the left sides of the DFN models) of the four cases are shown in Fig. 10. Due to the smaller flow rates in the DFN models considering Barton's e/E model,2 the breakthrough curves were less steep and
Fig. 10. Temperature break through curve (outflow borders at the left side of DFN models).
exhibited a long tail effect compared with the DFN models considering Li and Jiang's e/E model.27 From the temperature distributions in DFN models of different e/E model (Fig. 11), it was observed that thermal front moved more slowly in DFN models considering Barton's e/E model.2 Temperature distributions in each realization of Case 1 are compared in Fig. 12, and the comparison focuses on the same or different temperature distributions considering Barton's e/E
36
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
Fig. 11. Temperature distributions in DFN models with varying JRC distributions and e/E models (t ¼1 105 s).
Fig. 12. Temperature distributions in 5 realizations considering normal distribution of JRC (Case 1). (a) Considering the e/E model in Barton2 (t¼ 5 105 s). (b) Considering the e/E model in Li and Jiang27 (t ¼ 1 105 s).
model2 and Li and Jiang's e/E model,27 respectively. Due to the channeling (heterogeneity of flow rates, Fig. 9), the temperature distributions in each realization of Case 1 considering Barton's e/E model2 were slightly different. In contrast, the temperature distributions in each realization of Case 1 considering Li and Jiang's
e/E model27 were almost the same. This phenomenon was also reflected in the temperature breakthrough curves for different realizations of Case 1 (Fig. 13). Therefore, heat transfer in fracture networks is dependent on flow rates that are affected by the e/E models.
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
37
transfer in DFN models considering Li and Jiang's e/E model27 would be neglected for a first order approximation. However, Barton's e/E model 2 usually estimates small values of e/E ratio, so fracture surface roughness would play an important role in fluid flow and heat transfer in DFN models considering Barton's e/E model.2 In addition, Barton's e/E model2 is dependent on the mechanical apertures. Therefore, a proper e/E model should be chosen to accurately model the coupled fluid flow and heat transfer processes in fractured rocks. (3) Macroscopic fluid flow and heat transfer in different realizations of DFN models considering Barton's e/E model2 are different from each other, and thus a number of realizations are needed to reflect the fluid flow and heat transfer behavior in this case. If considering Li and Jiang's e/E model,27 one realization is sufficient to predict the fluid flow and heat transfer processes in DFN models. Lang et al.29 showed that two dimensional analysis cannot be used to approximately estimate the three dimensional equivalent permeability of fractured rock mass. Therefore, the present conclusions based on two dimensional simulations may or may not be extended to three dimensional case, and a systematic verification is needed. In this study, we considered the macroscopic fluid flow and heat transfer affected by fracture surface roughness. In practice, fracture roughness may vary due to other mechanical processes, and gouge materials may fill the fracture voids. Guo et al. 30 pointed out that thermal stress can also play an important significant role in flow pattern evolution and heat production. These factors can influence the fracture permeability and consequently the heat transfer and are important in the coupled THMC modeling, but they are beyond the scope of this paper. The relationships used in this study are mainly derived based on the equivalent fluid flow calculation, without considering the effects of tortuosity. This approximation may be valid for modeling of macroscopic fluid flow, but it may underestimate the thermal transfer area in fractures. To better understand in-situ problems, further studies on mechanical-hydraulic aperture relationship and roughness distribution are strongly recommended in the future.
Acknowledgments
Fig. 13. Temperature breakthrough curves for the 5 realizations considering normal distributions of JRC (Case 1). (a) Considering the e/E model in Barton.2 (b) Considering the e/E model in Li and Jiang.27
5. Conclusion and implication Based on the empirical relationships of mechanical-hydraulic apertures and the estimated distributions of fracture surface roughness (JRC), the role of fracture roughness distribution in macroscopic fracture flow and heat transfer through fractured rocks was numerically investigated. The results show that both flow rates and heat breakthrough curves can be affected to some extent by different mechanical-hydraulic aperture relationships and various roughness distributions. We can come to the main conclusions as follows: (1) Based on the JRC data in Oskarshamn/Forsmark and Bakhtiary, normal and lognormal distributions can be used to properly describe the JRC distributions in these two sites, respectively. DFN models with the normal distribution of JRC are less permeable than those with the lognormal distribution of JRC. As a result, the movement of thermal front is relatively slower in DFN models with the normal distribution of JRC. (2) Li and Jiang's e/E model27 predicts a relatively small influence of surface roughness (large values of e/E ratio), and thus the effects of fracture surface roughness on fluid flow and heat
This work was jointly supported by the National Natural Science Foundation of China (Nos. 41272279, 51322401 and 51509138), Beijing Natural Science Foundation (No. 8152020), the Recruitment Program of Beijing Youth Experts (2014000020124G115), the Openend Research Fund of the State Key Laboratory for Geomechanics and Deep Underground Engineering (No. SKLGDUEK1407), and the Scientific Research Foundation of the Education Ministry for Returned Chinese Scholars, China.
References 1. Grasselli G, Wirth J, Egger P. Quantitative three-dimensional description of a rough surface and parameter evolution with shearing. Int J Rock Mech Min Sci. 2002;39:789–800. 2. Barton N, Bandis S, Bakhtar K. Strength, deformation and conductivity coupling of rock joints. Int J Rock Mech Min Sci. 1985;22:121–140. 3. Myers N. Characterization of surface roughness. Wear. 1962;5:182–189. 4. Zimmerman RW, Bodvarsson GS. Hydraulic conductivity of rock fractures. Transp Porous Med. 1996;23:1–30. 5. Huang SL, Oelfke SM, Speck RC. Applicability of fractal characterization and modelling to rock joint profiles. Int J Rock Mech Min Sci. 1992;29:135–153. 6. Tatone BSA, Grasselli G. A new 2D discontinuity roughness parameter and its correlation with JRC. Int J Rock Mech Min Sci. 2010;47:1391–1400. 7. Tse R, Cruden DM. Estimating joint roughness coefficients. Int J Rock Mech Min Sci Geomech Abstr. 1979;16:303–307. 8. Yu X, Vayssade B. Joint profiles and their roughness parameters. Int J Rock Mech Min Sci Geomech Abstr. 1991;28:333–336.
38
S. Luo et al. / International Journal of Rock Mechanics & Mining Sciences 87 (2016) 29–38
9. Gao Y, Wong LNY. A modified correlation between roughness parameter Z2 and the JRC. Rock Mech Rock Eng. 2015;48:387–396. 10. Li Y, Huang R. Relationship between joint roughness coefficient and fractal dimension of rock fracture surfaces. Int J Rock Mech Min Sci. 2015;75:15–22. 11. Boutt DF, Grasselli G, Fredrich JT, Cook BK, Williams JR. Trapping zones: the effect of fracture roughness on the directional anisotropy of fluid flow and colloid transport in a single fracture. Geophys Res Lett. 2006;33:L21402. 12. Min KB, Jing L, Stephansson O. Fracture system characterization and evaluation of the equivalent permeability tensor of fractured rock masses using a stochastic REV approach. Hydrogeol J. 2004;12:497–510. 13. Shaik AR, Rahman SS, Tran NH, Tran T. Numerical simulation of fluid-rock coupling heat transfer in naturally fractured geothermal system. Appl Therm Eng. 2011;31:1600–1606. 14. Zhao Z, Jing L, Neretnieks I, Moreno L. Numerical modeling of stress effects on solute transport in fractured rocks. Comput Geotech. 2011;38:113–126. 15. Barton N. Review of a new shear-strength criterion for rock joints. Eng Geol. 1973;7:287–332. 16. Barton N, Choubey V. The shear strength of rock joints in theory and practice. Rock Mech. 1977;10:1–54. 17. Asadollahi P. Stability Analysis of a Single Three Dimensional Rock Block: Effect of Dilatancy and High-Velocity Water Jet Impact [PhD Thesis]. Austin: UTAU; 2009. 18. Sanei M, Faramarzi L, Fahimifar A, Goli S, Mehinrad A, Rahmati A. Shear strength of discontinuities in sedimentary rock masses based on direct shear tests. Int J Rock Mech Min Sci. 2015;75:119–131. 19. Tsang YW, Witherspoon PA. The dependence of fracture mechanical and fluid flow properties on fracture roughness and sample size. J Geophys Res. 1983;88:2359–2366. 20. Zhao Z, Li B. On the role of fracture surface roughness in fluid flow and solute transport through fractured rocks. In: Proceeding of the 13th International
Congress of Rock Mechanics. Montreal, Canada; 10–13 May 2015. 21. Patir N, Cheng HS. An average flow model for determining effects of threedimensional roughness on partial hydrodynamic lubrication. J Lubric Tech. 1978;100:12–17. 22. Renshaw CE. On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J Geophys Res. 1995;100:24629–24636. 23. Waite M, Ge S, Spetzler H. A new conceptual model for flow in discrete fractures: An experimental and numerical study. J Geophys Res. 1999;104:13049– 13059. 24. Olsson R, Barton N. An improved model for hydromechanical coupling during shearing of rock joints. Int J Rock Mech Min Sci. 2001;38:317–329. 25. Rasouli V, Hosseinian A. Correlations developed for estimation of hydraulic parameters of rough fractures through the simulation of JRC flow channels. Rock Mech Rock Eng. 2011;44:447–461. 26. Xiong X, Li B, Jiang Y, Koyama T, Zhang C. Experimental and numerical study of the geometrical and hydraulic characteristics of a single rock fracture during shear. Int J Rock Mech Min Sci. 2011;48:1292–1302. 27. Li B, Jiang Y. Quantitative estimation of fluid flow mechanism in rock fracture taking into account the influences of JRC and Reynolds number. J MMIJ. 2013;129:479–484 [in Japanese]. 28. Fu P, Hao Y, Walsh SDC, Carrigan CR. Thermal drawdown-induced flow channeling in fracture geothermal Reservoirs. Rock Mech Rock Eng. 2016;49:1001– 1024. 29. Guo B, Fu P, Hao Y, Peters CA, Carrigan CR. Thermal drawdown-induced flow channeling in a single fracture in EGS. Geothermics. 2016;61:46–62. 30. Lang P, Paluszny A, Zimmerman R. Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J Geophys Res. 2014;119:6288–6307.