Non-linear flow behaviour of rough fractures having standard JRC profiles

Technical Note

Non-linear?ow behaviour of rough fractures having standard

JRC pro?les

Mahdi Zoorabadi n,Serkan Saydam,Wendy Timms,Bruce Hebblewhite

School of Mining Engineering,UNSW Australia,Sydney2052,NSW,Australia

a r t i c l e i n f o

Article history:

Received28July2014

Received in revised form

10November2014

Accepted4March2015

Available online2April2015

Keywords:

Rock fracture

Nonlinear?ow conditions

JRC pro?les

1.Introduction

The Darcian?ow law,which expresses a linear relationship

between water pressure gradient and?ow rate,is commonly used

for interpretation of hydrogeological?eld tests[1,2].In jointed

rocks where the transmissivity is typically controlled by existing

fractures,non-linear?ow conditions can sometimes occur.The

non-linear relationship between water pressure gradient and?ow

rate may have considerable effects on the transmissivity calculated

from?eld tests[3–9].

The validity of the Darcy’s law and linear?ow expressions for

analysing water?ow within rock fractures have been widely

studied.Louis[10]empirically de?ned?ve steady state?ow

regimes for rock fractures.These?ow regimes start from laminar,

through to fully developed turbulent?ow depending on relative

roughness and?ow velocity.In fact,switching from linear Darcy

?ow to turbulent?ow is not immediate,and there is a transitional

regime between them[11].

Reynolds number(Re)is commonly used for distinguishing

between linear?ow(laminar)and nonlinear?ow regions(tran-

sient and turbulent?ow).It is a dimensionless number that gives a

measure of the ratio of inertial forces to viscous forces[12,13]:

Re?eρV2T=

μv

e

?VD e=?e1T

whereρis the density of water,V is the mean velocity of water,μis

the dynamic viscosity of water,D e is the hydraulic diameter of the

fracture(equal to two times the hydraulic aperture),and?is the

kinematic viscosity of water.A wide range of Reynolds number has

been reported in the literature for distinguishing between linear

and nonlinear?ow conditions.Romm[14]conducted laboratory

tests on fractures with different surface roughness and found that

the Darcy’s law is valid when the Reynolds number is smaller than

2400.A Reynolds number less than500–600was reported by

Louis[10]for smooth surfaces to have a linear?ow condition.

Durham and Bonner[15],Skjetne et al.[16],and Zimmerman and

Yeo[17]concluded that the transition from linear laminar?ow to

nonlinear?ow occurred at Reynolds numbers between1and25.

Wittke[18]and Lee and Farmer[19]suggested that transition

to nonlinearity occurs in the range100–2300for Reynolds

number.Although these studies had been done at different rough-

ness geometry and different?ow channel,the reported threshold

Reynolds numbers cover a wide range.

For single-phase?ow through a given rock joint with width of

w,the transmissivity has the following relationship with?ow rate

(Q)and pressure gradiente?pT:

T?

àμQ

w?p

e2T

Since the ratio of?ow rate and pressure gradient is constant for

the linear?ow condition(Darcy?ow),the transmissivity for linear

?ow is a constant value and can be calculated by the following

equation,known as cubic law:

T0?

a h3

e3T

Contents lists available at ScienceDirect

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International Journal of

Rock Mechanics&Mining Sciences

https://www.360docs.net/doc/336659730.html,/10.1016/j.ijrmms.2015.03.004

1365-1609/&2015Elsevier Ltd.All rights

reserved.

n Corresponding author.

E-mail address:m.zoorabadi@https://www.360docs.net/doc/336659730.html,.au(M.Zoorabadi).

International Journal of Rock Mechanics&Mining Sciences76(2015)192–199

where a h is the hydraulic aperture of the rough rock joint.The relationship between pressure gradient and ?ow rate for a nonlinear ?ow regime is commonly presented by the following two equations which were introduced by Forchheimer [20],and Izbash [21],respectively:à?p ?aQ tbQ 2e4Tà?p ?λQ m

e5T

where a and b ,are constants for the Forchheimer equation,which represent the linear and nonlinear components of pressure drop.Based on Darcy ’s law,the linear component can be calculated as a ?μ=T 0where μis the dynamic viscosity of water.The constant b in the Forchheimer equation and the constants,λand m ,in Izbash ’s equation are determined by back analysis of ?ow tests.Zimmerman et al.[22]further conducted laboratory experiments along with numerical modelling to investigate nonlinear ?ow through rough fractures.They plotted the normalised transmissiv-ity,the ratio between measured transmissivity (T )and that calculated from linear ?ow (T 0),against Reynolds number.They showed that for Reynolds numbers higher than 10,the nonlinear-ity increases rapidly.Nonlinear ?ow behaviour can be modelled by the Forchheimer equation [20,22]as follows:T ?

T 01tβR e

e6T

where βis a dimensionless constant and can be found by curve ?tting techniques.Al-Yaarubi [23]stated that the critical Reynolds number (threshold for linear ?ow regime)decreases with increas-ing fracture

roughness.

Fig.1.Arti ?cial joints corresponding to JRC pro ?les:(a)(8–10),(b)(10–12),(c)(12–14),(d)(14–16),(e)(16–18),(e)(18–

20).

Fig.2.Jig to adjust inlet gap for ?ow test on rough surfaces.

M.Zoorabadi et al./International Journal of Rock Mechanics &Mining Sciences 76(2015)192–199

193

Ranjith and Darlington [24]conducted laboratory experiments on single-phase ?ow through rough fractures.Their results con-?rmed that a Reynolds number of 10was a good approximation for distinguishing between the linear ?ow and nonlinear ?ow (Forchheimer regime)behaviour.Zhang and Nemcik [25]studied the ?ow regimes in laboratory experiments on tension fractures.Their study included ?ow tests through rough fractures under different applied normal stresses.They found that the analytical Forchheimer formulation and the empirical Izbash ’s equation have similar capabilities for modelling the nonlinear ?ow behaviour in rough fractures.

In the present paper,nonlinear ?ow through rough fractures is studied in laboratory experiments on some standard roughness pro ?les.The laboratory tests were completed by ?owing water through two matched Joint Roughness Coef ?cient (JRC)pro ?les.This study investigates the ?ow regimes for JRC pro ?les and explores the relationship between pressure gradient and ?ow rate for these pro ?les beyond the range of previously tested conditions by using advanced experimental techniques.

2.Test set-up and procedure

The joint roughness coef ?cient,JRC,which is a tool for quantifying roughness of real rock joints,was originally intro-duced by Barton and Choubey [26].In most cases,the roughness of the rock joints at the laboratory scale (10cm)is determined by the comparison of the joint ’s roughness pro ?les with proposed JRC pro ?les.These two-dimensional pro ?les were digitised by Tatone [27]with a sampling interval of approximately 0.5mm.Since the

standard JRC pro ?les were originally obtained using a pro ?le comb from laboratory samples with an accuracy of 0.5to 1mm,the digitised coordinates have proven to be suf ?ciently accurate [27].The digitised information showed that the maximum asperity amplitudes for JRC pro ?les 0–2,2–4,4–6,and 6–8are in the range of (0.6–1.71mm).By considering the resolution and the accuracy of measuring systems,arti ?cial joints of JRC pro ?les 8–10,10–12,12–14,14–16,16–18,and 18–20were prepared for the laboratory experiments.

In this regard,an advanced cutting technique called Electric Discharge Machining (EDM)was used to cut the real geometry of JRC pro ?les on aluminium blocks.Two-dimensional JRC pro ?les were extruded in depth to have a rough surface of 10?10cm dimension (Fig.1).In the EDM technique,the material is removed from the workpiece by a series of rapidly recurring current dis-charges between two electrodes,separated by a dielectric liquid and subject to an electric voltage [28].This method provided an accuracy of 0.1mm for the cutting of asperities on pro ?les.After cutting the pro ?les,their surfaces were mapped using an optical pro ?le projector.The comparison of the mapped surfaces with digitised points of the JRC pro ?les showed that the Root-Mean-Square Error (RMSE)between samples and their digitised pro ?les was about 0.034mm.Therefore,the calculated RMSE represents a high level of accuracy for this cutting method.

A custom designed jig was developed to adjust the inlet gap with high accuracy (0.05mm).This system provides the ability to do ?ow tests through JRC pro ?les or other rough surfaces with a 10?10cm dimension (Fig.2).

For investigation of the hydraulic properties of JRC pro ?les under linear and nonlinear ?ow conditions,water ?ow tests were conducted with different opening (mechanical aperture)and different water heads for each sample.The test set-up also inclu-ded an upper water tank,inlet and outlet reservoirs and a lower tank (as shown in Fig.3a and b).The two faces of the JRC pro ?les were sealed by silicon to provide the hydraulic

connection

https://www.360docs.net/doc/336659730.html,boratory experiment set-up for water ?ow test on JRC pro ?les;c is after Al-Johar [32]

.

Fig.4.Normalised transmissivity plotted against Re for JRC pro ?les,Forchheimer equation with βvalues found by Zimmerman et al.[22].

Table 1

Z 2values for tested JRC pro ?les [27].JRC pro ?le Z 28–100.19310–120.21212–140.23814–160.27716–180.32318–20

0.395

M.Zoorabadi et al./International Journal of Rock Mechanics &Mining Sciences 76(2015)192–199

194

Fig.5.3D scattering of normalised transmissivity against Reynolds number and relative

roughness.

Fig.6.Pressure gradient against ?ow rate,and curve ?tting of Forchheimer and Izbash equations (JRC 8–10and 10–12).

M.Zoorabadi et al./International Journal of Rock Mechanics &Mining Sciences 76(2015)192–199195

between the inlet and outlet reservoir only through the joint (schematically shown in Fig.3c).

3.Results and discussion

For each JRC pro ?le,?ow tests were conducted for three mechanical apertures (opening between upper and lower sur-faces)and different pressure gradients (112?ow tests).Although the designed set up has capability to apply a wide range of water pressure gradient,for this study it limited to pressure gradient of 0.05–2.In Fig.4,the normalised transmissivity was plotted against Reynolds number.The Forchheimer equation (Eq.5)was success-fully ?tted to the results with two βvalues which were introduced by Zimmerman et al.[22].This ?gure shows that the Forchheimer equation with β?0:00477can represent the trend satisfactorily.It can be seen from Fig.4that the normalised transmissivity decreases with increasing roughness.The variation of Reynolds number from 10to 300reduces the normalised transmissivity from 0.95to 0.4.

Lomize [29]and Louis [10]showed that the roughness geome-try controls the hydraulic properties of rough fractures.They used

the expression of Relative Roughness which is common for ?ow through pipes and channels as an index for quantifying the roughness geometry.This parameter represents the effect of roughness on ?ow behaviour,which decreases with an increase of the gap between the upper and lower surfaces of rough fractures.The roughness of rock fractures and JRC pro ?les are different with the roughness inside a pipe.The JRC pro ?les consist of several irregularities and their roughness cannot be shown only by asperity amplitude.The water ?ow within JRC pro ?les is affected by small-scale irregularities and undulations.Tse and Cruden [30]used Z 2as a tool to measure the fracture roughness.This parameter represents the root mean square of the ?rst derivative of the asperities ’amplitudes and is calculated as follows:

Z 2?1Z L x ?0dz

2????????????????????????????????????????????????

1n eΔx TX n i ?1eZ i t1àZ i T2

v u u t e7Twhere L is the total length of the pro ?le and Z i t1,Z i are the

elevations of two successive points over the length of the pro ?le,n is the total number of points,and Δx is the interval between two successive points.Tatone [27]calculated Z 2for JRC pro ?les

and

Fig.7.Pressure gradient against ?ow rate,and curve ?tting of Forchheimer and Izbash equations (JRC 12–14and 14–16).

M.Zoorabadi et al./International Journal of Rock Mechanics &Mining Sciences 76(2015)192–199

196

their values for the tested pro ?les are listed in Table 1.In fact,Z 2represents the variation of asperity height along a given direction.

In this paper,the relative roughness is de ?ned as the ratio between Z 2and the hydraulic aperture.Zoorabadi et al.[31]applied a semi-analytical procedure on JRC pro ?les and found the following relationship between aperture ratio (hydraulic aperture to mechanical aperture)and JRC value,a h =a m ?0:9912à4:653e à6?JRC 3:303

e8T

This equation is based on the cubic law and is valid for linear ?ow conditions.In Fig.5,the scattering plot of normalised against Reynolds number and relative roughness is shown.The higher relative roughness represents a fracture with a higher roughness and smaller opening.The scattering on the plot of the normalised transmissivity against Reynolds number can be explained using the results presented in Fig.5.It can be seen that for a relative roughness between 0.6and 1.4the effective Reynolds number is limited to Re

less

Fig.8.Pressure gradient against ?ow rate,and curve ?tting of Forchheimer and Izbash equations (JRC 16–18and 18–

20).

Fig.9.Variation of constant parameter m with relative roughness (Izabsh

Equation).Fig.10.Variation of constant parameter λwith relative roughness (Izabsh Equation).

M.Zoorabadi et al./International Journal of Rock Mechanics &Mining Sciences 76(2015)192–199197

than 100.The effective range of Reynolds numbers would be in the range of 10–900for a relative roughness between 0.2and 0.4.These ranges are limited to the applied pressure gradient.

The variation of measured transmissivity with Reynolds num-ber and relative roughness (Figs.4and 5)shows a non-constant ratio between ?ow rate and pressure gradient (nonlinear ?ow conditions)in Eq.(4).In Figs.6–8,the applied pressure gradient is plotted against measured and predicted ?ow rates for each ?ow test.The cubic law has been used to calculate the predicted ?ow rates.These results show a clear nonlinearity between the pres-sure gradient and ?ow rate for the JRC pro ?les.The Forchheimer and Izbash equations (Eqs.(6)and (4))were ?tted to the results to compare their ability to represent this nonlinear conditions.The regression analysis demonstrates the capability of both equations to model the nonlinear relationship between pressure gradient and ?ow rate for rough fractures.

The constant parameters of both equations have been deter-mined by regression analysis for each JRC pro ?le with different relative roughness.For the Izbash equation,the variation range of m is between 1.1and 1.39with an average of 1.234and standard deviation of 0.071.This parameter does not indicate a clear corr-elation with relative roughness (Fig.9).Unlike m the constant parameter of λincreases clearly with relative roughness increasing (Fig.10).For the Forchheimer equation,both constant parameters (a,b )show an increasing trend with relative roughness (Figs.11and 12).The relative roughness increases with increasing of fracture roughness (increasing JRC)for same range of fracture aperture.Then these results con ?rm that the nonlinearity is higher in the fractures with higher roughness.

4.Conclusions

Nonlinear ?ow conditions can occur inside rock fractures during hydrogeological tests such as packer tests and well tests,particularly with high ?ow rates.However,the assumption of linear ?ow conditions for interpretation of these tests can sig-ni ?cantly affect the results and underestimate the hydraulic

conductivity.This paper presents a detailed investigation of non-linear ?ow through rough fractures,quantifying the impact of nonlinear conditions on the relationship between hydraulic gra-dient and transmissivity.For this study,?ow tests were conducted on JRC pro ?les which are commonly used to quantify fracture roughness.

This study shows that the Reynolds number and relative rough-ness control nonlinear ?ow within rough fractures.The results from this study are consistent with previous studies [22,24]which showed that the normalised transmissivity decreases with increas-ing Reynolds number.The range of this reducing trend was limited to Re less than 100.This range has been signi ?cantly extended by this study which provides a reference for most range of applica-tions.The effects of relative roughness on the transmissivity of rough fractures were investigated by 3D plots of normalised transmissivity against Reynolds number and relative roughness.The relative roughness controls the effective range of the Reynolds number.For a higher range of relative roughness which represents smaller aperture or higher roughness,the in ?uence of the Reynolds number is limited.

The nonlinear relationship between pressure gradient and ?ow rate has been studied for rough fractures.The Izbash and For-chheimer equations were ?tted to ?ow test results for each JRC pro ?le.Both equations were demonstrated to have a similar ability to model nonlinear ?ow conditions.For the ?rst time,the empirical constants of Izbash and Forchheimer equations were developed for each JRC pro ?les.It was shown that the variation range of constant parameter of m in the Izbash equation is independent of relative roughness.These equations has potential to be used in the numerical modelling of water ?ow through fractured rocks to increase the accuracy of ?ow modelling.

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