research article fractal analysis of permeability of ... · to develop a mechanistic model for...
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Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 490320 5 pageshttpdxdoiorg1011552013490320
Research ArticleFractal Analysis of Permeability of Unsaturated Fractured Rocks
Guoping Jiang1 Wei Shi2 and Lili Huang2
1 Earthquake Engineering Research Test Center Guangzhou University Guangzhou 510405 China2Ningbo Polytechnic Ningbo Zhejiang 315800 China
Correspondence should be addressed to Guoping Jiang lp2002999126com
Received 12 February 2013 Accepted 4 March 2013
Academic Editors A Billi and A V Koustov
Copyright copy 2013 Guoping Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A physical conceptual model for water retention in fractured rocks is derived while taking into account the effect of pore sizedistribution and tortuosity of capillaries The formula of calculating relative hydraulic conductivity of fractured rock is given basedon fractal theory It is an issue to choose an appropriate capillary pressure-saturation curve in the research of unsaturated fracturedmass The geometric pattern of the fracture bulk is described based on the fractal distribution of tortuosity The resulting watercontent expression is then used to estimate the unsaturated hydraulic conductivity of the fractured medium based on the well-knownmodel of Burdine It is found that for large enough ranges of fracture apertures the new constitutive model converges to theempirical Brooks-Corey model
1 Introduction
Modeling water flow in unsaturated fractured rocks hasreceived considerable attention in the last two decades Oneof themain reasons for focussing on the study of liquid flow inthis type of media is that spontaneous capillary imbibition isan important fundamental phenomenon existing extensivelyin a variety of processes such as oil recovery polymer com-posite manufacturing soil science and hydrology The otherreason is that deep disposal in crystalline rocks is consideredto be an effective mean of isolating radioactive wastes fromthe biosphere The study of basic transport processes haslong been recognized because of which one can heightenthe comprehensive understanding of physical phenomenasuch as permeability [1 2] heat transfer [3ndash5] and sorption[6 7] Many parameters such as the porosity size of poreand tortuosity of capillaries are very important for fluidflow in hard rocks These parameters however are closelyrelated to the geometric architecture of hard rocks Cai et alhave analyzed the natural fractured trace maps representinga wide variety of scales geological settings and lithologies[8] Cai and Yu have reported the density of different sizedfault segments within the San Andreas fault zone which isfractal [9]The distributions of fracture aperture and fracturespacing are self-similar over a well-defined range of apertures
in the Cajon Pass scientific drill hole which was found byBarton and Zoback [10] Based on the assumption that thefracture pattern is self-similar the Sierpinski carpet was oftenemployed to simulate porous media [11] The conceptualconstitutive model proposed by Wu and Yu [12] had usedits fractal dimension to the parameters of the Brooks-Coreyconstitutive model [13] through the Sierpinski carpet ASierpinski space was also adopted to characterize the spatialdistribution of a drainage network in the Gardon basinFrance [14]
From the above review it is shown that a mechanisticmodel has not yet been established In this paper we attemptto develop a mechanistic model for unsaturated flow infractured hard rocks based on the method using the specificfractal to describe fractured rock The expressions of theproposed constitutive model are closed form and easy toevaluate Another important feature the tortuosity fractaldimension 119863
119879 which affected other model parameters and
should not be neglected while it was often neglected in thepast investigation is considered
Now fractal theory has a wide variety of applicationsin sciences and engineering fields such as thermal science[15ndash19] fluid science [1 2 6] and industrial constructionengineering [20] For example Moussa [2] has systematicallyinvestigated the transport of porous media based on fractal
2 The Scientific World Journal
theory The work [2] is open Xiao et al [3ndash5 15 16 1819] have done much outstanding work on heat transfer offluids by using fractal technique In our work we derive theanalytical expressions for the relative hydraulic conductivityof fractured rock while taking into account the effect of poresize distribution based on the fractal geometry theory
2 Construction of the Fractal Model
Themodel is presented in Figure 1 It has been shown that thecumulative size distribution of contact spots on engineeringsurfaces is similar to islands on earth andpores in porous rockwhich follows the fractal scaling law [2]
119873(119871 ge 119863119899) = (
119863119899max
119863119899
)
119889119891
(1)
where 119889119891is the fractal dimension for pores 119863 is diameter
119871 is the length scale and 119873 is the total number of poreswhose sizes equal to and greater than 119863
119899 The number of
pores whose sizes range from119863119899to119863 + 119889119863
119899is
minus119889119873 = 119889119891119863119889119891
119899max119863minus(119889119891+1)
119899 119889119863119899
(2)
when water flow through the pores of porous rock thecapillaries may be tortuous These tortuous capillaries couldbe expressed by fractal equation [21] Consider
119871119886(119863119899) = 119871119863119879
01198631minus119863119879
119899 (3)
where 119863119879is the tortuosity fractal dimension and lies in the
range 1 lt 119863119879lt 2 which represents the extent of convoluted
ness of capillary pathways for fluid flow through a mediumNote that for a straight capillary path 119863
119879= 1 and a higher
value of 119863119879corresponds to a highly tortuous capillary Let
the diameter of a capillary in the medium be 119863119899and let its
tortuous length along the flow direction be 119871119886(119863119899) 1198710is
representative length of channels With a straight capillary119871119886(119863119899) = 1198710 The total volume of pores from119863
119899min to119863119899maxcan be obtained from (1) as
Vtol = int119863119899max
119863119899min
120587
41198632
119899119871119886 (minus119889119873)
= int
119863119899max
119863119899min
120587
41198632
119899119871119886119889119891119863119889119891
119899sdotmax119863minus(119889119891+1)
119899 119889119863119899
=120587
4 (3 minus 119889119891minus 119863119879)
119871119863119879
01198891198911198633minus119863119879
119899sdotmax
times (1 minus (119863119899min
119863119899max
)
(3minus119889119891minus119863119879)
)
(4)
where Vtol is the whole volume of the pores (minus119889119873) is givenby (2)
1198710119860
Figure 1 The model presented
Similarity to the state above the volume V(119863119899min119909lt119863119899)
ofporesparticles from 119863
119899to 119863119899max can be obtained from (1)
asV(119863119899minlt119863119899)
=120587
4119871119863119879
0119889119891119863119889119891
119899sdotmax int119863119899
119863119899min119909
1198632minus119889119891minus119863119879
119899 119889119863119899
=120587
4 (3 minus 119889119891minus 119863119879)
119871119863119879
0119889119891119863119889119891
119899sdotmax (1198633minus119889119891minus119863119879
119899 )100381610038161003816100381610038161003816
119863119899
119863119899min
(5)
The effective saturation 119878 of volume V(119863119899min119909lt119863119899)
can beobtained Consider
119878 =
V(119863119899minlt119863119899)
Vtol
=
100381610038161003816100381610038161003816
119863119899max119863119899min
(1198633minus119889119891minus119863119879
119899 )
100381610038161003816100381610038161003816
119863119899
119863119899min(1198633minus119889119891minus119863119879
119899 )
=1198633minus119889119891minus119863119879
119899 minus 1198633minus119889119891minus119863119879
119899min
1198633minus119889119891minus119863119879
119899max minus 1198633minus119889119891minus119863119879
119899min
(6)
Assuming the immiscible fluid flow in reservoir rocksthree relatively important forces are considered the capillarypressure can be expressed Consider
119901 =2120590cos (120573)120588119892119863119899
(7)
where 120590 is surface tension between the wetting and non-wetting fluids 120588 is the water density 119892 is the gravityacceleration 119863
119899is diameter of a pore and 120573 is contact angle
between the extraneous water and solidThe relative permeabilities are usually expressed in terms
of water saturation 119878119908 The saturation curve for the proposed
model of fractured rock was derived Consider
119878119908=1199013minus119889119891minus119863119879
119888 minus 1199013minus119889119891minus119863119879
119888min
1199013minus119889119891minus119863119879
119888min minus 1199013minus119889119891minus119863119879
119888max
(8)
The Scientific World Journal 3
119901119888
119878 0
(B-C model)(10E minus7 cm)
(10119890minus5 cm)(10Eminus3 cm)
1 10 100 1000 10000
001
01
1
(a) Effective saturation
119901119888
(B-C model)(10Eminus7cm)
(10Eminus5cm)(10Eminus3cm)
1119864minus12
1119864minus9
1119864minus6
1119864minus3
119870119863(119878)
1 10 100 1000 10000
1
(b) Relative hydraulic conductivity
Figure 2 Comparison between the Brooks-Corey model and the new relations for different ranges of fracture apertures when 119863119879= 1
where
119901119888max =
2120590cos (120573)120588119892119863119899min
119901119888min119909 =
2120590cos (120573)120588119892119863119899max
(9)
3 The Permeability of UnsaturatedFractured Rocks
The Burdine and Mualem models are the two most widelyused models predicting relative hydraulic conductivity Forthe particular state of flow in fractured hard rock the Burdinemodel seems to be more consistent and to be often adoptedThe expression of the Burdine model is
119870119863 (119904) = 119904
2int119904
0(119889119904119901
2
119888)
int1
0(1198891199041199012
119888)
(10)
where 119870119863(119904) is the relative hydraulic conductivity Inserting
(8) to (10) we obtain the following form for 119870119863(119904)
119870119863 (119904) = (
119901119889119891+119863119879minus3
119888 minus 119901119889119891+119863119879minus3
119888max
119901119889119891+119863119879minus3
119888min119909 minus 119901119889119891+119863119879minus3
119888max
)
2
times119901119889119891+119863119879minus5
119888 minus 119901119889119891+119863119879minus5
119888max
119901119889119891+119863119879minus5
119888min119909 minus 119901119889119891+119863119879minus5
119888max
(11)
The expressions of (6) and (10) represent the proposedconstitutive model for fractured hard rocks Note that allmodel parameters are determined by geometric parametersof119863119899max 119863119899min119909 and residual water content 120573The novel constitutive model has some similarities
with the well-known Brooks-Corey model which is 1198780=
(119901119888119901119889)minus120582 and 119870
119863(119904) = 119878
3+1205822 where 119901119889is the reciprocal of
air entry pressure and 120582 is a model parameter related to poresize distribution
When 119863119879= 1 120582 = 2 minus 119889
119891 and 119901
119888max ≪ 119901119888min 119909 119901119888max
is neglected here the model derived here is simplified to theBrooks-Corey model Comparisons of the proposed and theBrooks-Corey models for three different ranges of fractureapertures are depicted in Figure 2 The assumed parametersare 119889119891= 18 119863
119899max = 10minus1 cm and 119863
119899min119909 = 10minus3 cm
10minus5 cm and 10minus7 cm According to Figure 2 the Brooks-
Corey model seems to be adequate to describe the hydraulicproperties of fractured rocks for large ranges of fractureapertures and low values of pressure head
The relationship between the fractal dimension and effec-tive saturation is very important in the study of unsaturatedflow in fractured hard rocks The geometric parameters andphysical constants used for the analysis are 120573 = 0 120590 =
7225 dypa In order to analyze the influence of the fracturedensity we consider different values of 119863
119879 With (8) and
(11) the relationship is obtained (Figure 3) With the physicalconstants 119863
119899= 01 cm 119863
119899sdotmax = 1 cm and 119863119899sdotmin119909 =
001 cm the relationship between 119889119891and 119878
0can also be
determined (Figure 4)In order to test the proposed invasion depth model it is
crucial to correctly determine the fractal dimensions119863119879and
119889119891According to the following formula proved recently by Yu
and Li [22 23]
120601 = (119863119899min
119863119899max
)
119863119864minus119889119891
(12)
4 The Scientific World Journal
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100
1
119901119888
119870119863(119878)
DT = 1DT = 11DT = 115
(a) 119863119899min = 10minus3 cm
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100 1000 10000
DT = 1
DT = 11
DT = 115
119901119888
1
119870119863(119878)
(b) 119863119899min = 10minus7 cm
Figure 3 The new relations for 119870119863(119878) and119863
119879
where 119863119864is the Euclidean dimension and 119863
119864= 2 is used
in this work The maximum pore diameter can be calculatedbased on the model of square arrangement of particles
According to the following formula proved recently byJiangchao Cai where119870 is permeability 119896 is Kozeny constant
1 12 14 16 180
005
01
015
02
025
03
035
04
DT = 1
DT = 102
DT = 104
DT = 106
119878 0
119889119891
Figure 4 The new relations for 1198780and 119889
119891
which considers the tortuosity of capillaries and pore non-uniformity And the parameters 119870 and 119896 are all proved [23]Consider
119863119879= 1 +
ln 119905avln 1198710119905av
119863119899max =
2 (1 minus 120601)
120601radic119870119896
120601(radic
120601
1 minus 120601+ radic
120587
4 (1 minus 120601)minus 1)
(13)
where
119905av =1
2
[[[
[
1 +1
2radic1 minus 120601 +
radic(radic1 minus 120601 minus 1)2
+ ((1 minus 120601) 4)
1 minus radic1 minus 120601
]]]
]
1198710
119905av=
119889119891minus 1
2radic1 minus 120601
120601
120587
119889119891(2 minus 119889
119891)
119863119899min
119863119899max
(14)
From (12) and (13) the relationship among 120601 119863119879 and 119889
119891is
determined (Figure 5)
4 Conclusions
In this paper with the consideration of pore size distributionand tortuosity of capillaries a new fractal model for relativehydraulic conductivity of fractured rock is developed Thederived constitutive model is an effort to understand andcharacterize unsaturated flow in fractured rocks The expres-sions of water content and relative hydraulic conductivitycurves have analytical closed forms The parameters canbe completely determined by the geometry of the fractalmodel Every parameter of the proposed formulas of calcu-lating relative hydraulic conductivity of fractured rock has
The Scientific World Journal 5
0 02 04 06 08 1
1
12
14
16
18
2
119863119879
119889119891
120593
119863119899min119909119863119899max = 0001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 0001
Figure 5 The relationship among 120601119863119879 and 119889
119891
a clear physical meaning The fractal model can reveal themechanisms of hydraulic conductivity for fluids flow throughunsaturated rocksThe tortuosity fractal dimension119863T whichaffected other model parameters and should not be neglectedwhile it was often neglected in the past investigation isconsidered in this paper
References
[1] B Berkowitz and A Hadad ldquoFractal and multifractal measuresof natural and synthetic fracture networksrdquo Journal of Geophys-ical Research B vol 102 no 6 pp 12205ndash12218 1997
[2] R Moussa ldquoIs the drainage network a fractal SierpinskispacerdquoWater Resources Research vol 33 pp 2399ndash2408 1997
[3] BM Yu J C Cai andMQ Zou ldquoOn the physical properties ofapparent two-phase fractal porousmediardquoVadose Zone Journalvol 8 no 1 pp 177ndash186 2009
[4] B Q Xiao J T Fan and F Ding ldquoPrediction of relativepermeability of unsaturated porous media based on fractaltheory and monte carlo simulationrdquo Energy amp Fuels vol 26 pp6971ndash6978 2012
[5] B Q Xiao B M Yu Z Wang and L Chen ldquoA fractal modelfor heat transfer of nanofluids by convection in a poolrdquo PhysicsLetters A vol 373 no 45 pp 4178ndash4181 2009
[6] B Q Xiao and B M Yu ldquoA fractal analysis of subcooled flowboiling heat transferrdquo International Journal of Multiphase Flowvol 33 no 10 pp 1126ndash1139 2007
[7] P G Obuko and K Aki ldquoFractal geometry in the San Andreasfault systemrdquo Journal of Geophysical Research vol 92 pp 345ndash355 1987
[8] J C Cai B M YuM Q Zou andM FMei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010
[9] J C Cai and B M Yu ldquoPrediction of maximum pore size ofporous media based on fractal geometryrdquo Fractals vol 18 no4 pp 417ndash423 2010
[10] C A Barton and M D Zoback ldquoSelf-similar distribution andproperties of macroscopic fractures at depth in crystalline rockin the Cajon Pass scientific drill holerdquo Journal of GeophysicalResearch vol 97 no 4 pp 5181ndash5200 1992
[11] SW Tyler and SWWheatcraft ldquoFractal processes in soil waterretentionrdquo Water Resources Research vol 26 no 5 pp 1047ndash1054 1990
[12] J S Wu and B M Yu ldquoA fractal resistance model for flowthrough porous mediardquo International Journal of Heat and MassTransfer pp 3925ndash3932 2007
[13] R H Brooks and A T Corey Hydraulic Properties of PorousMedia Hydrology Paper no 3 lorado State University FortCollins CO 1964
[14] N T Burdine ldquoRelative permeability from pore size dis-tribution datardquo Transactions American Institute of MiningMetallurgical and Petroleum Engineers vol 198 pp 71ndash78 1953
[15] B Q Xiao ldquoPrediction of heat transfer of nanofluid on criticalheat flux based on fractal geometryrdquo Chinese Physics B vol 22no 1 Article ID 014402 2013
[16] B Q Xiao ldquoA new analytical model for heat transfer in poolboilingrdquoModern Physics Letters B vol 24 no 12 pp 1229ndash12362010
[17] T M Nordberg and S T Thorolfsson ldquoLow impact develop-ment and bioretention areas in cold climatesrdquo in Proceedings oftheWorldWater and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 3409ndash3418 Salt Lake City Utah USA July 2004
[18] M B Green J R Martin and P Griffin ldquoTreatment ofcombined sewer overflows at small wastewater treatment worksby constructed reed bedsrdquo Water Science and Technology vol40 no 3 pp 357ndash364 1999
[19] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008
[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and formationrdquo Physical ReviewLetters vol 54 pp 1325ndash1328 1985
[21] W F Hunt and A R Jarrett ldquoEvaluating bioretention areasfrom two field sites in North Carolinardquo in Proceedings of theWorld Water and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 797ndash806 Salt Lake City Utah USA July 2004
[22] BM Yu and J H Li ldquoSome fractal characters of porousmediardquoFractals vol 9 no 3 pp 365ndash372 2001
[23] B M Yu and HM Li ldquoA geometry model for tortuosity of flowpath in porous mediardquo Chinese Physics Letters vol 21 no 8 pp1569ndash1571 2004
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2 The Scientific World Journal
theory The work [2] is open Xiao et al [3ndash5 15 16 1819] have done much outstanding work on heat transfer offluids by using fractal technique In our work we derive theanalytical expressions for the relative hydraulic conductivityof fractured rock while taking into account the effect of poresize distribution based on the fractal geometry theory
2 Construction of the Fractal Model
Themodel is presented in Figure 1 It has been shown that thecumulative size distribution of contact spots on engineeringsurfaces is similar to islands on earth andpores in porous rockwhich follows the fractal scaling law [2]
119873(119871 ge 119863119899) = (
119863119899max
119863119899
)
119889119891
(1)
where 119889119891is the fractal dimension for pores 119863 is diameter
119871 is the length scale and 119873 is the total number of poreswhose sizes equal to and greater than 119863
119899 The number of
pores whose sizes range from119863119899to119863 + 119889119863
119899is
minus119889119873 = 119889119891119863119889119891
119899max119863minus(119889119891+1)
119899 119889119863119899
(2)
when water flow through the pores of porous rock thecapillaries may be tortuous These tortuous capillaries couldbe expressed by fractal equation [21] Consider
119871119886(119863119899) = 119871119863119879
01198631minus119863119879
119899 (3)
where 119863119879is the tortuosity fractal dimension and lies in the
range 1 lt 119863119879lt 2 which represents the extent of convoluted
ness of capillary pathways for fluid flow through a mediumNote that for a straight capillary path 119863
119879= 1 and a higher
value of 119863119879corresponds to a highly tortuous capillary Let
the diameter of a capillary in the medium be 119863119899and let its
tortuous length along the flow direction be 119871119886(119863119899) 1198710is
representative length of channels With a straight capillary119871119886(119863119899) = 1198710 The total volume of pores from119863
119899min to119863119899maxcan be obtained from (1) as
Vtol = int119863119899max
119863119899min
120587
41198632
119899119871119886 (minus119889119873)
= int
119863119899max
119863119899min
120587
41198632
119899119871119886119889119891119863119889119891
119899sdotmax119863minus(119889119891+1)
119899 119889119863119899
=120587
4 (3 minus 119889119891minus 119863119879)
119871119863119879
01198891198911198633minus119863119879
119899sdotmax
times (1 minus (119863119899min
119863119899max
)
(3minus119889119891minus119863119879)
)
(4)
where Vtol is the whole volume of the pores (minus119889119873) is givenby (2)
1198710119860
Figure 1 The model presented
Similarity to the state above the volume V(119863119899min119909lt119863119899)
ofporesparticles from 119863
119899to 119863119899max can be obtained from (1)
asV(119863119899minlt119863119899)
=120587
4119871119863119879
0119889119891119863119889119891
119899sdotmax int119863119899
119863119899min119909
1198632minus119889119891minus119863119879
119899 119889119863119899
=120587
4 (3 minus 119889119891minus 119863119879)
119871119863119879
0119889119891119863119889119891
119899sdotmax (1198633minus119889119891minus119863119879
119899 )100381610038161003816100381610038161003816
119863119899
119863119899min
(5)
The effective saturation 119878 of volume V(119863119899min119909lt119863119899)
can beobtained Consider
119878 =
V(119863119899minlt119863119899)
Vtol
=
100381610038161003816100381610038161003816
119863119899max119863119899min
(1198633minus119889119891minus119863119879
119899 )
100381610038161003816100381610038161003816
119863119899
119863119899min(1198633minus119889119891minus119863119879
119899 )
=1198633minus119889119891minus119863119879
119899 minus 1198633minus119889119891minus119863119879
119899min
1198633minus119889119891minus119863119879
119899max minus 1198633minus119889119891minus119863119879
119899min
(6)
Assuming the immiscible fluid flow in reservoir rocksthree relatively important forces are considered the capillarypressure can be expressed Consider
119901 =2120590cos (120573)120588119892119863119899
(7)
where 120590 is surface tension between the wetting and non-wetting fluids 120588 is the water density 119892 is the gravityacceleration 119863
119899is diameter of a pore and 120573 is contact angle
between the extraneous water and solidThe relative permeabilities are usually expressed in terms
of water saturation 119878119908 The saturation curve for the proposed
model of fractured rock was derived Consider
119878119908=1199013minus119889119891minus119863119879
119888 minus 1199013minus119889119891minus119863119879
119888min
1199013minus119889119891minus119863119879
119888min minus 1199013minus119889119891minus119863119879
119888max
(8)
The Scientific World Journal 3
119901119888
119878 0
(B-C model)(10E minus7 cm)
(10119890minus5 cm)(10Eminus3 cm)
1 10 100 1000 10000
001
01
1
(a) Effective saturation
119901119888
(B-C model)(10Eminus7cm)
(10Eminus5cm)(10Eminus3cm)
1119864minus12
1119864minus9
1119864minus6
1119864minus3
119870119863(119878)
1 10 100 1000 10000
1
(b) Relative hydraulic conductivity
Figure 2 Comparison between the Brooks-Corey model and the new relations for different ranges of fracture apertures when 119863119879= 1
where
119901119888max =
2120590cos (120573)120588119892119863119899min
119901119888min119909 =
2120590cos (120573)120588119892119863119899max
(9)
3 The Permeability of UnsaturatedFractured Rocks
The Burdine and Mualem models are the two most widelyused models predicting relative hydraulic conductivity Forthe particular state of flow in fractured hard rock the Burdinemodel seems to be more consistent and to be often adoptedThe expression of the Burdine model is
119870119863 (119904) = 119904
2int119904
0(119889119904119901
2
119888)
int1
0(1198891199041199012
119888)
(10)
where 119870119863(119904) is the relative hydraulic conductivity Inserting
(8) to (10) we obtain the following form for 119870119863(119904)
119870119863 (119904) = (
119901119889119891+119863119879minus3
119888 minus 119901119889119891+119863119879minus3
119888max
119901119889119891+119863119879minus3
119888min119909 minus 119901119889119891+119863119879minus3
119888max
)
2
times119901119889119891+119863119879minus5
119888 minus 119901119889119891+119863119879minus5
119888max
119901119889119891+119863119879minus5
119888min119909 minus 119901119889119891+119863119879minus5
119888max
(11)
The expressions of (6) and (10) represent the proposedconstitutive model for fractured hard rocks Note that allmodel parameters are determined by geometric parametersof119863119899max 119863119899min119909 and residual water content 120573The novel constitutive model has some similarities
with the well-known Brooks-Corey model which is 1198780=
(119901119888119901119889)minus120582 and 119870
119863(119904) = 119878
3+1205822 where 119901119889is the reciprocal of
air entry pressure and 120582 is a model parameter related to poresize distribution
When 119863119879= 1 120582 = 2 minus 119889
119891 and 119901
119888max ≪ 119901119888min 119909 119901119888max
is neglected here the model derived here is simplified to theBrooks-Corey model Comparisons of the proposed and theBrooks-Corey models for three different ranges of fractureapertures are depicted in Figure 2 The assumed parametersare 119889119891= 18 119863
119899max = 10minus1 cm and 119863
119899min119909 = 10minus3 cm
10minus5 cm and 10minus7 cm According to Figure 2 the Brooks-
Corey model seems to be adequate to describe the hydraulicproperties of fractured rocks for large ranges of fractureapertures and low values of pressure head
The relationship between the fractal dimension and effec-tive saturation is very important in the study of unsaturatedflow in fractured hard rocks The geometric parameters andphysical constants used for the analysis are 120573 = 0 120590 =
7225 dypa In order to analyze the influence of the fracturedensity we consider different values of 119863
119879 With (8) and
(11) the relationship is obtained (Figure 3) With the physicalconstants 119863
119899= 01 cm 119863
119899sdotmax = 1 cm and 119863119899sdotmin119909 =
001 cm the relationship between 119889119891and 119878
0can also be
determined (Figure 4)In order to test the proposed invasion depth model it is
crucial to correctly determine the fractal dimensions119863119879and
119889119891According to the following formula proved recently by Yu
and Li [22 23]
120601 = (119863119899min
119863119899max
)
119863119864minus119889119891
(12)
4 The Scientific World Journal
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100
1
119901119888
119870119863(119878)
DT = 1DT = 11DT = 115
(a) 119863119899min = 10minus3 cm
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100 1000 10000
DT = 1
DT = 11
DT = 115
119901119888
1
119870119863(119878)
(b) 119863119899min = 10minus7 cm
Figure 3 The new relations for 119870119863(119878) and119863
119879
where 119863119864is the Euclidean dimension and 119863
119864= 2 is used
in this work The maximum pore diameter can be calculatedbased on the model of square arrangement of particles
According to the following formula proved recently byJiangchao Cai where119870 is permeability 119896 is Kozeny constant
1 12 14 16 180
005
01
015
02
025
03
035
04
DT = 1
DT = 102
DT = 104
DT = 106
119878 0
119889119891
Figure 4 The new relations for 1198780and 119889
119891
which considers the tortuosity of capillaries and pore non-uniformity And the parameters 119870 and 119896 are all proved [23]Consider
119863119879= 1 +
ln 119905avln 1198710119905av
119863119899max =
2 (1 minus 120601)
120601radic119870119896
120601(radic
120601
1 minus 120601+ radic
120587
4 (1 minus 120601)minus 1)
(13)
where
119905av =1
2
[[[
[
1 +1
2radic1 minus 120601 +
radic(radic1 minus 120601 minus 1)2
+ ((1 minus 120601) 4)
1 minus radic1 minus 120601
]]]
]
1198710
119905av=
119889119891minus 1
2radic1 minus 120601
120601
120587
119889119891(2 minus 119889
119891)
119863119899min
119863119899max
(14)
From (12) and (13) the relationship among 120601 119863119879 and 119889
119891is
determined (Figure 5)
4 Conclusions
In this paper with the consideration of pore size distributionand tortuosity of capillaries a new fractal model for relativehydraulic conductivity of fractured rock is developed Thederived constitutive model is an effort to understand andcharacterize unsaturated flow in fractured rocks The expres-sions of water content and relative hydraulic conductivitycurves have analytical closed forms The parameters canbe completely determined by the geometry of the fractalmodel Every parameter of the proposed formulas of calcu-lating relative hydraulic conductivity of fractured rock has
The Scientific World Journal 5
0 02 04 06 08 1
1
12
14
16
18
2
119863119879
119889119891
120593
119863119899min119909119863119899max = 0001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 0001
Figure 5 The relationship among 120601119863119879 and 119889
119891
a clear physical meaning The fractal model can reveal themechanisms of hydraulic conductivity for fluids flow throughunsaturated rocksThe tortuosity fractal dimension119863T whichaffected other model parameters and should not be neglectedwhile it was often neglected in the past investigation isconsidered in this paper
References
[1] B Berkowitz and A Hadad ldquoFractal and multifractal measuresof natural and synthetic fracture networksrdquo Journal of Geophys-ical Research B vol 102 no 6 pp 12205ndash12218 1997
[2] R Moussa ldquoIs the drainage network a fractal SierpinskispacerdquoWater Resources Research vol 33 pp 2399ndash2408 1997
[3] BM Yu J C Cai andMQ Zou ldquoOn the physical properties ofapparent two-phase fractal porousmediardquoVadose Zone Journalvol 8 no 1 pp 177ndash186 2009
[4] B Q Xiao J T Fan and F Ding ldquoPrediction of relativepermeability of unsaturated porous media based on fractaltheory and monte carlo simulationrdquo Energy amp Fuels vol 26 pp6971ndash6978 2012
[5] B Q Xiao B M Yu Z Wang and L Chen ldquoA fractal modelfor heat transfer of nanofluids by convection in a poolrdquo PhysicsLetters A vol 373 no 45 pp 4178ndash4181 2009
[6] B Q Xiao and B M Yu ldquoA fractal analysis of subcooled flowboiling heat transferrdquo International Journal of Multiphase Flowvol 33 no 10 pp 1126ndash1139 2007
[7] P G Obuko and K Aki ldquoFractal geometry in the San Andreasfault systemrdquo Journal of Geophysical Research vol 92 pp 345ndash355 1987
[8] J C Cai B M YuM Q Zou andM FMei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010
[9] J C Cai and B M Yu ldquoPrediction of maximum pore size ofporous media based on fractal geometryrdquo Fractals vol 18 no4 pp 417ndash423 2010
[10] C A Barton and M D Zoback ldquoSelf-similar distribution andproperties of macroscopic fractures at depth in crystalline rockin the Cajon Pass scientific drill holerdquo Journal of GeophysicalResearch vol 97 no 4 pp 5181ndash5200 1992
[11] SW Tyler and SWWheatcraft ldquoFractal processes in soil waterretentionrdquo Water Resources Research vol 26 no 5 pp 1047ndash1054 1990
[12] J S Wu and B M Yu ldquoA fractal resistance model for flowthrough porous mediardquo International Journal of Heat and MassTransfer pp 3925ndash3932 2007
[13] R H Brooks and A T Corey Hydraulic Properties of PorousMedia Hydrology Paper no 3 lorado State University FortCollins CO 1964
[14] N T Burdine ldquoRelative permeability from pore size dis-tribution datardquo Transactions American Institute of MiningMetallurgical and Petroleum Engineers vol 198 pp 71ndash78 1953
[15] B Q Xiao ldquoPrediction of heat transfer of nanofluid on criticalheat flux based on fractal geometryrdquo Chinese Physics B vol 22no 1 Article ID 014402 2013
[16] B Q Xiao ldquoA new analytical model for heat transfer in poolboilingrdquoModern Physics Letters B vol 24 no 12 pp 1229ndash12362010
[17] T M Nordberg and S T Thorolfsson ldquoLow impact develop-ment and bioretention areas in cold climatesrdquo in Proceedings oftheWorldWater and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 3409ndash3418 Salt Lake City Utah USA July 2004
[18] M B Green J R Martin and P Griffin ldquoTreatment ofcombined sewer overflows at small wastewater treatment worksby constructed reed bedsrdquo Water Science and Technology vol40 no 3 pp 357ndash364 1999
[19] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008
[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and formationrdquo Physical ReviewLetters vol 54 pp 1325ndash1328 1985
[21] W F Hunt and A R Jarrett ldquoEvaluating bioretention areasfrom two field sites in North Carolinardquo in Proceedings of theWorld Water and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 797ndash806 Salt Lake City Utah USA July 2004
[22] BM Yu and J H Li ldquoSome fractal characters of porousmediardquoFractals vol 9 no 3 pp 365ndash372 2001
[23] B M Yu and HM Li ldquoA geometry model for tortuosity of flowpath in porous mediardquo Chinese Physics Letters vol 21 no 8 pp1569ndash1571 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
The Scientific World Journal 3
119901119888
119878 0
(B-C model)(10E minus7 cm)
(10119890minus5 cm)(10Eminus3 cm)
1 10 100 1000 10000
001
01
1
(a) Effective saturation
119901119888
(B-C model)(10Eminus7cm)
(10Eminus5cm)(10Eminus3cm)
1119864minus12
1119864minus9
1119864minus6
1119864minus3
119870119863(119878)
1 10 100 1000 10000
1
(b) Relative hydraulic conductivity
Figure 2 Comparison between the Brooks-Corey model and the new relations for different ranges of fracture apertures when 119863119879= 1
where
119901119888max =
2120590cos (120573)120588119892119863119899min
119901119888min119909 =
2120590cos (120573)120588119892119863119899max
(9)
3 The Permeability of UnsaturatedFractured Rocks
The Burdine and Mualem models are the two most widelyused models predicting relative hydraulic conductivity Forthe particular state of flow in fractured hard rock the Burdinemodel seems to be more consistent and to be often adoptedThe expression of the Burdine model is
119870119863 (119904) = 119904
2int119904
0(119889119904119901
2
119888)
int1
0(1198891199041199012
119888)
(10)
where 119870119863(119904) is the relative hydraulic conductivity Inserting
(8) to (10) we obtain the following form for 119870119863(119904)
119870119863 (119904) = (
119901119889119891+119863119879minus3
119888 minus 119901119889119891+119863119879minus3
119888max
119901119889119891+119863119879minus3
119888min119909 minus 119901119889119891+119863119879minus3
119888max
)
2
times119901119889119891+119863119879minus5
119888 minus 119901119889119891+119863119879minus5
119888max
119901119889119891+119863119879minus5
119888min119909 minus 119901119889119891+119863119879minus5
119888max
(11)
The expressions of (6) and (10) represent the proposedconstitutive model for fractured hard rocks Note that allmodel parameters are determined by geometric parametersof119863119899max 119863119899min119909 and residual water content 120573The novel constitutive model has some similarities
with the well-known Brooks-Corey model which is 1198780=
(119901119888119901119889)minus120582 and 119870
119863(119904) = 119878
3+1205822 where 119901119889is the reciprocal of
air entry pressure and 120582 is a model parameter related to poresize distribution
When 119863119879= 1 120582 = 2 minus 119889
119891 and 119901
119888max ≪ 119901119888min 119909 119901119888max
is neglected here the model derived here is simplified to theBrooks-Corey model Comparisons of the proposed and theBrooks-Corey models for three different ranges of fractureapertures are depicted in Figure 2 The assumed parametersare 119889119891= 18 119863
119899max = 10minus1 cm and 119863
119899min119909 = 10minus3 cm
10minus5 cm and 10minus7 cm According to Figure 2 the Brooks-
Corey model seems to be adequate to describe the hydraulicproperties of fractured rocks for large ranges of fractureapertures and low values of pressure head
The relationship between the fractal dimension and effec-tive saturation is very important in the study of unsaturatedflow in fractured hard rocks The geometric parameters andphysical constants used for the analysis are 120573 = 0 120590 =
7225 dypa In order to analyze the influence of the fracturedensity we consider different values of 119863
119879 With (8) and
(11) the relationship is obtained (Figure 3) With the physicalconstants 119863
119899= 01 cm 119863
119899sdotmax = 1 cm and 119863119899sdotmin119909 =
001 cm the relationship between 119889119891and 119878
0can also be
determined (Figure 4)In order to test the proposed invasion depth model it is
crucial to correctly determine the fractal dimensions119863119879and
119889119891According to the following formula proved recently by Yu
and Li [22 23]
120601 = (119863119899min
119863119899max
)
119863119864minus119889119891
(12)
4 The Scientific World Journal
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100
1
119901119888
119870119863(119878)
DT = 1DT = 11DT = 115
(a) 119863119899min = 10minus3 cm
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100 1000 10000
DT = 1
DT = 11
DT = 115
119901119888
1
119870119863(119878)
(b) 119863119899min = 10minus7 cm
Figure 3 The new relations for 119870119863(119878) and119863
119879
where 119863119864is the Euclidean dimension and 119863
119864= 2 is used
in this work The maximum pore diameter can be calculatedbased on the model of square arrangement of particles
According to the following formula proved recently byJiangchao Cai where119870 is permeability 119896 is Kozeny constant
1 12 14 16 180
005
01
015
02
025
03
035
04
DT = 1
DT = 102
DT = 104
DT = 106
119878 0
119889119891
Figure 4 The new relations for 1198780and 119889
119891
which considers the tortuosity of capillaries and pore non-uniformity And the parameters 119870 and 119896 are all proved [23]Consider
119863119879= 1 +
ln 119905avln 1198710119905av
119863119899max =
2 (1 minus 120601)
120601radic119870119896
120601(radic
120601
1 minus 120601+ radic
120587
4 (1 minus 120601)minus 1)
(13)
where
119905av =1
2
[[[
[
1 +1
2radic1 minus 120601 +
radic(radic1 minus 120601 minus 1)2
+ ((1 minus 120601) 4)
1 minus radic1 minus 120601
]]]
]
1198710
119905av=
119889119891minus 1
2radic1 minus 120601
120601
120587
119889119891(2 minus 119889
119891)
119863119899min
119863119899max
(14)
From (12) and (13) the relationship among 120601 119863119879 and 119889
119891is
determined (Figure 5)
4 Conclusions
In this paper with the consideration of pore size distributionand tortuosity of capillaries a new fractal model for relativehydraulic conductivity of fractured rock is developed Thederived constitutive model is an effort to understand andcharacterize unsaturated flow in fractured rocks The expres-sions of water content and relative hydraulic conductivitycurves have analytical closed forms The parameters canbe completely determined by the geometry of the fractalmodel Every parameter of the proposed formulas of calcu-lating relative hydraulic conductivity of fractured rock has
The Scientific World Journal 5
0 02 04 06 08 1
1
12
14
16
18
2
119863119879
119889119891
120593
119863119899min119909119863119899max = 0001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 0001
Figure 5 The relationship among 120601119863119879 and 119889
119891
a clear physical meaning The fractal model can reveal themechanisms of hydraulic conductivity for fluids flow throughunsaturated rocksThe tortuosity fractal dimension119863T whichaffected other model parameters and should not be neglectedwhile it was often neglected in the past investigation isconsidered in this paper
References
[1] B Berkowitz and A Hadad ldquoFractal and multifractal measuresof natural and synthetic fracture networksrdquo Journal of Geophys-ical Research B vol 102 no 6 pp 12205ndash12218 1997
[2] R Moussa ldquoIs the drainage network a fractal SierpinskispacerdquoWater Resources Research vol 33 pp 2399ndash2408 1997
[3] BM Yu J C Cai andMQ Zou ldquoOn the physical properties ofapparent two-phase fractal porousmediardquoVadose Zone Journalvol 8 no 1 pp 177ndash186 2009
[4] B Q Xiao J T Fan and F Ding ldquoPrediction of relativepermeability of unsaturated porous media based on fractaltheory and monte carlo simulationrdquo Energy amp Fuels vol 26 pp6971ndash6978 2012
[5] B Q Xiao B M Yu Z Wang and L Chen ldquoA fractal modelfor heat transfer of nanofluids by convection in a poolrdquo PhysicsLetters A vol 373 no 45 pp 4178ndash4181 2009
[6] B Q Xiao and B M Yu ldquoA fractal analysis of subcooled flowboiling heat transferrdquo International Journal of Multiphase Flowvol 33 no 10 pp 1126ndash1139 2007
[7] P G Obuko and K Aki ldquoFractal geometry in the San Andreasfault systemrdquo Journal of Geophysical Research vol 92 pp 345ndash355 1987
[8] J C Cai B M YuM Q Zou andM FMei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010
[9] J C Cai and B M Yu ldquoPrediction of maximum pore size ofporous media based on fractal geometryrdquo Fractals vol 18 no4 pp 417ndash423 2010
[10] C A Barton and M D Zoback ldquoSelf-similar distribution andproperties of macroscopic fractures at depth in crystalline rockin the Cajon Pass scientific drill holerdquo Journal of GeophysicalResearch vol 97 no 4 pp 5181ndash5200 1992
[11] SW Tyler and SWWheatcraft ldquoFractal processes in soil waterretentionrdquo Water Resources Research vol 26 no 5 pp 1047ndash1054 1990
[12] J S Wu and B M Yu ldquoA fractal resistance model for flowthrough porous mediardquo International Journal of Heat and MassTransfer pp 3925ndash3932 2007
[13] R H Brooks and A T Corey Hydraulic Properties of PorousMedia Hydrology Paper no 3 lorado State University FortCollins CO 1964
[14] N T Burdine ldquoRelative permeability from pore size dis-tribution datardquo Transactions American Institute of MiningMetallurgical and Petroleum Engineers vol 198 pp 71ndash78 1953
[15] B Q Xiao ldquoPrediction of heat transfer of nanofluid on criticalheat flux based on fractal geometryrdquo Chinese Physics B vol 22no 1 Article ID 014402 2013
[16] B Q Xiao ldquoA new analytical model for heat transfer in poolboilingrdquoModern Physics Letters B vol 24 no 12 pp 1229ndash12362010
[17] T M Nordberg and S T Thorolfsson ldquoLow impact develop-ment and bioretention areas in cold climatesrdquo in Proceedings oftheWorldWater and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 3409ndash3418 Salt Lake City Utah USA July 2004
[18] M B Green J R Martin and P Griffin ldquoTreatment ofcombined sewer overflows at small wastewater treatment worksby constructed reed bedsrdquo Water Science and Technology vol40 no 3 pp 357ndash364 1999
[19] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008
[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and formationrdquo Physical ReviewLetters vol 54 pp 1325ndash1328 1985
[21] W F Hunt and A R Jarrett ldquoEvaluating bioretention areasfrom two field sites in North Carolinardquo in Proceedings of theWorld Water and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 797ndash806 Salt Lake City Utah USA July 2004
[22] BM Yu and J H Li ldquoSome fractal characters of porousmediardquoFractals vol 9 no 3 pp 365ndash372 2001
[23] B M Yu and HM Li ldquoA geometry model for tortuosity of flowpath in porous mediardquo Chinese Physics Letters vol 21 no 8 pp1569ndash1571 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
4 The Scientific World Journal
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100
1
119901119888
119870119863(119878)
DT = 1DT = 11DT = 115
(a) 119863119899min = 10minus3 cm
1119864minus12
1119864minus9
1119864minus6
1119864minus3
1 10 100 1000 10000
DT = 1
DT = 11
DT = 115
119901119888
1
119870119863(119878)
(b) 119863119899min = 10minus7 cm
Figure 3 The new relations for 119870119863(119878) and119863
119879
where 119863119864is the Euclidean dimension and 119863
119864= 2 is used
in this work The maximum pore diameter can be calculatedbased on the model of square arrangement of particles
According to the following formula proved recently byJiangchao Cai where119870 is permeability 119896 is Kozeny constant
1 12 14 16 180
005
01
015
02
025
03
035
04
DT = 1
DT = 102
DT = 104
DT = 106
119878 0
119889119891
Figure 4 The new relations for 1198780and 119889
119891
which considers the tortuosity of capillaries and pore non-uniformity And the parameters 119870 and 119896 are all proved [23]Consider
119863119879= 1 +
ln 119905avln 1198710119905av
119863119899max =
2 (1 minus 120601)
120601radic119870119896
120601(radic
120601
1 minus 120601+ radic
120587
4 (1 minus 120601)minus 1)
(13)
where
119905av =1
2
[[[
[
1 +1
2radic1 minus 120601 +
radic(radic1 minus 120601 minus 1)2
+ ((1 minus 120601) 4)
1 minus radic1 minus 120601
]]]
]
1198710
119905av=
119889119891minus 1
2radic1 minus 120601
120601
120587
119889119891(2 minus 119889
119891)
119863119899min
119863119899max
(14)
From (12) and (13) the relationship among 120601 119863119879 and 119889
119891is
determined (Figure 5)
4 Conclusions
In this paper with the consideration of pore size distributionand tortuosity of capillaries a new fractal model for relativehydraulic conductivity of fractured rock is developed Thederived constitutive model is an effort to understand andcharacterize unsaturated flow in fractured rocks The expres-sions of water content and relative hydraulic conductivitycurves have analytical closed forms The parameters canbe completely determined by the geometry of the fractalmodel Every parameter of the proposed formulas of calcu-lating relative hydraulic conductivity of fractured rock has
The Scientific World Journal 5
0 02 04 06 08 1
1
12
14
16
18
2
119863119879
119889119891
120593
119863119899min119909119863119899max = 0001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 0001
Figure 5 The relationship among 120601119863119879 and 119889
119891
a clear physical meaning The fractal model can reveal themechanisms of hydraulic conductivity for fluids flow throughunsaturated rocksThe tortuosity fractal dimension119863T whichaffected other model parameters and should not be neglectedwhile it was often neglected in the past investigation isconsidered in this paper
References
[1] B Berkowitz and A Hadad ldquoFractal and multifractal measuresof natural and synthetic fracture networksrdquo Journal of Geophys-ical Research B vol 102 no 6 pp 12205ndash12218 1997
[2] R Moussa ldquoIs the drainage network a fractal SierpinskispacerdquoWater Resources Research vol 33 pp 2399ndash2408 1997
[3] BM Yu J C Cai andMQ Zou ldquoOn the physical properties ofapparent two-phase fractal porousmediardquoVadose Zone Journalvol 8 no 1 pp 177ndash186 2009
[4] B Q Xiao J T Fan and F Ding ldquoPrediction of relativepermeability of unsaturated porous media based on fractaltheory and monte carlo simulationrdquo Energy amp Fuels vol 26 pp6971ndash6978 2012
[5] B Q Xiao B M Yu Z Wang and L Chen ldquoA fractal modelfor heat transfer of nanofluids by convection in a poolrdquo PhysicsLetters A vol 373 no 45 pp 4178ndash4181 2009
[6] B Q Xiao and B M Yu ldquoA fractal analysis of subcooled flowboiling heat transferrdquo International Journal of Multiphase Flowvol 33 no 10 pp 1126ndash1139 2007
[7] P G Obuko and K Aki ldquoFractal geometry in the San Andreasfault systemrdquo Journal of Geophysical Research vol 92 pp 345ndash355 1987
[8] J C Cai B M YuM Q Zou andM FMei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010
[9] J C Cai and B M Yu ldquoPrediction of maximum pore size ofporous media based on fractal geometryrdquo Fractals vol 18 no4 pp 417ndash423 2010
[10] C A Barton and M D Zoback ldquoSelf-similar distribution andproperties of macroscopic fractures at depth in crystalline rockin the Cajon Pass scientific drill holerdquo Journal of GeophysicalResearch vol 97 no 4 pp 5181ndash5200 1992
[11] SW Tyler and SWWheatcraft ldquoFractal processes in soil waterretentionrdquo Water Resources Research vol 26 no 5 pp 1047ndash1054 1990
[12] J S Wu and B M Yu ldquoA fractal resistance model for flowthrough porous mediardquo International Journal of Heat and MassTransfer pp 3925ndash3932 2007
[13] R H Brooks and A T Corey Hydraulic Properties of PorousMedia Hydrology Paper no 3 lorado State University FortCollins CO 1964
[14] N T Burdine ldquoRelative permeability from pore size dis-tribution datardquo Transactions American Institute of MiningMetallurgical and Petroleum Engineers vol 198 pp 71ndash78 1953
[15] B Q Xiao ldquoPrediction of heat transfer of nanofluid on criticalheat flux based on fractal geometryrdquo Chinese Physics B vol 22no 1 Article ID 014402 2013
[16] B Q Xiao ldquoA new analytical model for heat transfer in poolboilingrdquoModern Physics Letters B vol 24 no 12 pp 1229ndash12362010
[17] T M Nordberg and S T Thorolfsson ldquoLow impact develop-ment and bioretention areas in cold climatesrdquo in Proceedings oftheWorldWater and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 3409ndash3418 Salt Lake City Utah USA July 2004
[18] M B Green J R Martin and P Griffin ldquoTreatment ofcombined sewer overflows at small wastewater treatment worksby constructed reed bedsrdquo Water Science and Technology vol40 no 3 pp 357ndash364 1999
[19] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008
[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and formationrdquo Physical ReviewLetters vol 54 pp 1325ndash1328 1985
[21] W F Hunt and A R Jarrett ldquoEvaluating bioretention areasfrom two field sites in North Carolinardquo in Proceedings of theWorld Water and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 797ndash806 Salt Lake City Utah USA July 2004
[22] BM Yu and J H Li ldquoSome fractal characters of porousmediardquoFractals vol 9 no 3 pp 365ndash372 2001
[23] B M Yu and HM Li ldquoA geometry model for tortuosity of flowpath in porous mediardquo Chinese Physics Letters vol 21 no 8 pp1569ndash1571 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
The Scientific World Journal 5
0 02 04 06 08 1
1
12
14
16
18
2
119863119879
119889119891
120593
119863119899min119909119863119899max = 0001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 001
119863119899min119909119863119899max = 0001
Figure 5 The relationship among 120601119863119879 and 119889
119891
a clear physical meaning The fractal model can reveal themechanisms of hydraulic conductivity for fluids flow throughunsaturated rocksThe tortuosity fractal dimension119863T whichaffected other model parameters and should not be neglectedwhile it was often neglected in the past investigation isconsidered in this paper
References
[1] B Berkowitz and A Hadad ldquoFractal and multifractal measuresof natural and synthetic fracture networksrdquo Journal of Geophys-ical Research B vol 102 no 6 pp 12205ndash12218 1997
[2] R Moussa ldquoIs the drainage network a fractal SierpinskispacerdquoWater Resources Research vol 33 pp 2399ndash2408 1997
[3] BM Yu J C Cai andMQ Zou ldquoOn the physical properties ofapparent two-phase fractal porousmediardquoVadose Zone Journalvol 8 no 1 pp 177ndash186 2009
[4] B Q Xiao J T Fan and F Ding ldquoPrediction of relativepermeability of unsaturated porous media based on fractaltheory and monte carlo simulationrdquo Energy amp Fuels vol 26 pp6971ndash6978 2012
[5] B Q Xiao B M Yu Z Wang and L Chen ldquoA fractal modelfor heat transfer of nanofluids by convection in a poolrdquo PhysicsLetters A vol 373 no 45 pp 4178ndash4181 2009
[6] B Q Xiao and B M Yu ldquoA fractal analysis of subcooled flowboiling heat transferrdquo International Journal of Multiphase Flowvol 33 no 10 pp 1126ndash1139 2007
[7] P G Obuko and K Aki ldquoFractal geometry in the San Andreasfault systemrdquo Journal of Geophysical Research vol 92 pp 345ndash355 1987
[8] J C Cai B M YuM Q Zou andM FMei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010
[9] J C Cai and B M Yu ldquoPrediction of maximum pore size ofporous media based on fractal geometryrdquo Fractals vol 18 no4 pp 417ndash423 2010
[10] C A Barton and M D Zoback ldquoSelf-similar distribution andproperties of macroscopic fractures at depth in crystalline rockin the Cajon Pass scientific drill holerdquo Journal of GeophysicalResearch vol 97 no 4 pp 5181ndash5200 1992
[11] SW Tyler and SWWheatcraft ldquoFractal processes in soil waterretentionrdquo Water Resources Research vol 26 no 5 pp 1047ndash1054 1990
[12] J S Wu and B M Yu ldquoA fractal resistance model for flowthrough porous mediardquo International Journal of Heat and MassTransfer pp 3925ndash3932 2007
[13] R H Brooks and A T Corey Hydraulic Properties of PorousMedia Hydrology Paper no 3 lorado State University FortCollins CO 1964
[14] N T Burdine ldquoRelative permeability from pore size dis-tribution datardquo Transactions American Institute of MiningMetallurgical and Petroleum Engineers vol 198 pp 71ndash78 1953
[15] B Q Xiao ldquoPrediction of heat transfer of nanofluid on criticalheat flux based on fractal geometryrdquo Chinese Physics B vol 22no 1 Article ID 014402 2013
[16] B Q Xiao ldquoA new analytical model for heat transfer in poolboilingrdquoModern Physics Letters B vol 24 no 12 pp 1229ndash12362010
[17] T M Nordberg and S T Thorolfsson ldquoLow impact develop-ment and bioretention areas in cold climatesrdquo in Proceedings oftheWorldWater and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 3409ndash3418 Salt Lake City Utah USA July 2004
[18] M B Green J R Martin and P Griffin ldquoTreatment ofcombined sewer overflows at small wastewater treatment worksby constructed reed bedsrdquo Water Science and Technology vol40 no 3 pp 357ndash364 1999
[19] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008
[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and formationrdquo Physical ReviewLetters vol 54 pp 1325ndash1328 1985
[21] W F Hunt and A R Jarrett ldquoEvaluating bioretention areasfrom two field sites in North Carolinardquo in Proceedings of theWorld Water and Environmental Resources Congress CriticalTransitions inWater and Environmental ResourcesManagementpp 797ndash806 Salt Lake City Utah USA July 2004
[22] BM Yu and J H Li ldquoSome fractal characters of porousmediardquoFractals vol 9 no 3 pp 365ndash372 2001
[23] B M Yu and HM Li ldquoA geometry model for tortuosity of flowpath in porous mediardquo Chinese Physics Letters vol 21 no 8 pp1569ndash1571 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in