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Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 429835 7 pageshttpdxdoiorg1011552013429835
Research ArticleNumerical Fractional-Calculus Model for Two-Phase Flow inFractured Media
Wenwen Zhong1 Changpin Li1 and Jisheng Kou2
1 Department of Mathematics Shanghai University Shanghai 200444 China2 School of Mathematics and Statistics Hubei Engineering University Xiaogan Hubei 432000 China
Correspondence should be addressed to Changpin Li lcpshueducn
Received 17 May 2013 Revised 13 July 2013 Accepted 13 July 2013
Academic Editor H Srivastava
Copyright copy 2013 Wenwen Zhong et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Numerical simulation of two-phase flow in fractured porous media is an important topic in the subsurface flow environmentalproblems and petroleum reservoir engineeringThe conventionalmodel does notworkwell inmany cases since it lacks thememoryproperty of fracture media In this paper we develop a new numerical formulation with fractional time derivative for two-phaseflow in fractured porous media In the proposed formulation the different fractional time derivatives are applied to fracture andmatrix regions since they have different memory properties We further develop a two-level time discrete method which uses alarge time step for the pressure and a small time step size for the saturationThe pressure equation is solved implicitly in each largetime step while the saturation is updated by an explicit fractional time scheme in each time substep Finally the numerical tests arecarried out to demonstrate the effectiveness of the proposed numerical model
1 Introduction
Numerical simulations for multiphase flow in fracturedporous media are very important in the subsurface flowenvironmental problems and petroleum reservoir engineer-ing Compared to usual heterogenous porous media thefractured media have two spaces with two distinct scalesthe fracture and the matrix The fractures possess the higherpermeability than the matrix but their volume is very smallwhen compared to the matrix Several conceptual models[1ndash11] have been developed to simulate the multiphase flowin porous media for example the single-porosity modelthe dual-porositydual-permeability model and the discrete-fracture model The review for these models can be found in[4]
These models can deal well with the two-phase flow withconstant physical parameters However the fluid flow mayperturb the porous formation by causing particle migrationresulting in pore clogging or chemically reacting with themedium to enlarge the pores or diminish the size of thepores [12] As a matter of fact the porosity and permeabilityoften depend on the fluid pressure and saturation For incom-pressible two-phase flow in petroleum reservoir engineering
the porosity varies small with respect to the pressure How-ever the injection fluids can change the porosity to a greatextent In the displacement process of oil by water theinjected water will make themedia wet and then the porositywill be enlarged along with the change of permeability Asa result the variability of the physical parameters shouldbe considered in modeling realistic two-phase flow Forflow in porous media Caputo [13] has studied the behaviorof fluxes in porous media using a memory formalism inwhich the ordinary time derivative is replaced by a fractionalderivative Garra [14] has studied a fractional time derivativegeneralization of a previous Natale-Salusti model aboutnonlinear temperature and pressure waves propagating influid-saturated porous rocks
In this paper we will study a fractional time derivativegeneralization of a classical two-phase flow model Theordinary time derivative is replaced by the Caputo fractionalderivative with variable lower limit of integral Because theCaputo fractional derivative is composed of the convolutionof a power law kernel and the ordinary derivative of thefunction it is a useful instrument to describe a powerlaw frequency variability of the physical coefficients Sincethe porosity and permeability have a remarkable contrast
2 Advances in Mathematical Physics
between matrixes and fractures the proposed formulationuses the different fractional time derivatives for the flowin fracture and matrix regions to represent their differentmemory properties
Many numerical methods have been developed in theliterature for example [15ndash17] In order to simulate thefractional two-phase flow we propose a two-level timediscrete method based on the physical property that thepressure varies less rapidly than the saturation [18] thatis a large time step is used for the pressure along with asmall time step size for the saturation The local fractionaldifferential equations have been widely studied in [19] Inour method the memory of saturations is restricted withineach large time step since the pressures are determined bysaturationswithout historicalmemoryTheproposedmethodis also viewed as the fractional generalization of the classicalIMplicit Pressure Explicit Saturation (IMPES) method [2021] which is a popular time-stepping approach employedin multiphase flow simulation Like IMPES we split thecoupled system into one pressure equation and one saturationequation based on the property of multiphysics processesof two-phase flow and treat the saturation and capillarypressure in the pressure equation explicitly to eliminate itsnonlinearity Each large time step is further divided into afew substeps in which the saturation is updated by an explicitfractional time scheme The cell-centered finite differencemethod [22] is employed for spatial discretization Finallynumerical results are given to demonstrate the validity of theproposed numerical model
2 Fractional Model of Two-PhaseIncompressible Flow
21 Fractional Model The Caputo fractional derivative isgiven by the convolution of a power law kernel and the ordi-nary derivative of the function So it is a useful instrument toconsider a power law frequency variability of the coefficientsby a simple convolution We now introduce the definition ofthe Caputo fractional derivative [17] as
119863120574
119886119905119910 (119905) =
1
Γ (1 minus 120574)int
119905
119886
(119905 minus 120591)minus1205741199101015840(120591) 119889120591 (1)
where 119910(119905) is a function and 0 lt 120574 lt 1 Here we use themodified Caputo fractional derivative that is the lower limitof integral in (1) is taken to be a function of 119905 instead of theconstant as usual that is 119886 = 119886(119905) The fractional derivativecan be used to describe the complex problems that involvememory in time because of the nonlocal property From this119886(119905) indicates the memory range at time 119905 and this memoryproperty varies with time
Using the Caputo fractional derivative we introducea memory formalism for two-phase incompressible andimmiscible fluid flow in porous media Denote the wettingphase by a subscript119908 and the nonwetting phase by 119899 Let 119878120572be the saturation of phase 120572 The two-phase saturations aresubject to the following constraint
119878119908 + 119878119899 = 1 (2)
For flow in porous media the velocity u120572 of each phase 120572is described by Darcyrsquos law as
u120572 = minus119896119903120572
120583120572
K (nabla119901120572 + 120588120572119892nabla119911) 120572 = 119908 119899 (3)
where K is the absolute permeability tensor in the porousmedium 119892 is the gravity acceleration 119911 is the depth and119896119903120572 120583120572 119901120572 and 120588120572 are the relative permeability viscositypressure and density of each phase respectively
Themass conservation equation of each phase is given by
120601119863120574
119886119905119878120572 + nabla sdot u120572 = 119902120572 120572 = 119908 119899 (4)
where 120601 is the porosity of the medium and 119902120572 is the externalmass flow rate
The difference between the nonwetting phase andwettingphase pressures is described by the capillary pressure
119901119888 (119878119908) = 119901119899 minus 119901119908 (5)
Denote 120582120572 = 119896119903120572120583120572 and Φ120572 = 119901120572 + 120588119892119911 Substituting (3)into (4) we obtain
120601119863120574
119886119905119878120572 + nabla sdot 120582120572KnablaΦ120572 = 119902120572 120572 = 119908 119899 (6)
Summing the forms (6) of two phases and taking into account(2) we can reach
minusnabla sdot 120582119905KnablaΦ119908 minus nabla sdot 120582119899KnablaΦ119888 = 119902119905 (7)
where 120582119905 = 120582119908 + 120582119899Φ119888 = 119901119888 + (120588119899 minus 120588119908)119892119911 and 119902119905 = 119902119908 + 119902119899Furthermore define u119886 = minus120582119905KnablaΦ119908 and 119891119908 = 120582119908120582119905 Wethen have
u119908 = 119891119908u119886 (8)
Thus the equation for the wetting-phase saturation becomes
120601119863120574
119886119905119878119908 + nabla sdot (119891119908u119886) = 119902119908 (9)
We note that the above two-phase flow formulation is afractional extension of the formulation [4] We consider theclassical mass conservative formulations of incompressibletwo-phase flow
120597120601119878120572
120597119905+ nabla sdot u120572 = 119902120572 120572 = 119908 119899 (10)
The porosity often depends on the fluid pressure and satu-ration In some problems of incompressible two-phase flowpetroleum reservoir engineering for example the porosityvaries small with respect to the pressure However theinjection fluids can change the porosity to a great extentIn the displacement process of oil by water the injectedwater will make the media wet and then the porosity willbe enlarged Summing the mass conservative formulations oftwo phases we obtain
120597120601
120597119905minus nabla sdot 120582119905KnablaΦ119908 minus nabla sdot 120582119899KnablaΦ119888 = 119902119905 (11)
Advances in Mathematical Physics 3
Since the porosity varies small with the pressure we assume120597120601120597119905 ≃ 0 in (11) and as a result a pressure equationis obtained with the same form to (9) The wetting-phasesaturation equation can be expressed as
120597120601119878119908
120597119905+ nabla sdot (119891119908u119886) = 119902119908 (12)
Because of porosity depending on 119878119908 we get by the chain rulethat
120597120601119878119908
120597119905= Υ (119878119908)
120597119878119908
120597119905 (13)
where Υ(119878119908) = 120601 + 1198781199081206011015840(119878119908) In general Υ(119878119908) can be
described by the power law With this argument we candeduce the fractional generalization of the classical two-phase model This clearly explains the physical reason to usethe fractional model
Finally we complete our model by the boundary andinitial conditions We divide the boundary 120597Ω of the com-putational domain Ω into the two nonoverlapping parts theDirichelt part Γ119863 and Newmann part Γ119873 where 120597Ω = Γ
119863cup
Γ119873 The pressure equation (6) is subject to the following
boundary conditions
119901119908 (or 119901119899) = 119901119863 on Γ119863
u120572 sdot n = u119873120572
on Γ119873(14)
where 119901119863 is the pressure on Γ119863 n is the outward unit normalvector to 120597Ω and u119873 is the imposed inflow rate on Γ119873 Theboundary conditions for the saturations are given by
119878119908 = 119878119873 on Γ119873 (15)
The initial saturation of the wetting phase is given by
119878119908 = 1198780
119908in Ω (16)
22 Discrete-Fracture Model with Fractional Time Deriva-tives Here the discrete-fracture model [4] is extended tothe case with the fractional time derivative The discrete-fracture model treats the matrix and fracture gridcells bydifferent geometrical dimensions that is if the domain is 119899-dimensional the matrix regions are n-dimensional but thefractures are simplified as the matrix gridcell interfaces thatare (119899 minus 1)-dimensional This treatment removes the length-scale contrast resulting from the explicit representation ofthe fracture aperture as in the single-porosity model so it iscapable to considerably improve the computational efficiencyand is convenient in practical implementation
We now decompose the entire domain into two parts thematrixΩ119898 and fractureΩ119891 The fractures are surrounded bythe matrix blocks We use the subscript 119898 to represent thematrix and the subscript 119891 to represent the fracture systemThe pressure in the matrix domain is determined by
minusnabla sdot 120582119905119898K119898nablaΦ119908119898 minus nabla sdot 120582119899119898K119898nablaΦ119888119898 = 119902119905119898 (17)
which is subject to the matrix-fracture interface condition
Φ119908119898 = Φ119908119891
Φ119888119898 = Φ119888119891 on 120597Ω119898 cap Ω119891(18)
In the fracture system we denote the fracture width by 120576and assume that the potentials are constant along the fracturewidth and then obtain the pressure equation in the fractureas
minusnabla sdot 120582119905119891K119891nablaΦ119908119891 minus nabla sdot 120582119899119891K119891nablaΦ119888119891 = 119902119905119891 + 119876119905119891 (19)
where 119876119905119891 is the mass transfer across the matrix-fractureinterfaces The above formulations are similar to the classicalmodel
As stated previously the fractional property representsthe variability of porosity and permeability which have aremarkable contrast between matrixes and fractures As aresult different fractional time derivatives should be usedfor the flow in the matrix regions and the fracture systemBy using the fractional time derivative we can express thesaturation equation in the matrix regions as
120601119898119863120574119898
119886119905119878119908119898 + nabla sdot (119891119908119898u119886119898) = 119902119908119898 (20)
along with the interface condition
119878119908119898 = 119878119908119891 on 120597Ω119898 cap Ω119891 (21)
Similarly the saturation equation in the fracture system isgiven by
120601119891119863120574119891
119886119905119878119908119891 + nabla sdot (119891119908119891u119886119891) = 119902119908119891 + 119876119908119891 (22)
where 119876119908119891 represents the mass transfer across the matrix-fracture interfaces
3 Numerical Methods
In this section we will present the numerical methods forthe fractional model of two-phase incompressible flow In thefollowing we focus on the time discretization schemes
We firstly divide the total time interval [0 119879] into 119873119901equal time steps as 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119873119901 = 119879 and denote thetime step length ℎ119901 = 119879119873119901This time division is used for thepressures Since the saturation varies more rapidly than thepressure we use a smaller time step size for saturation Eachsubinterval (119905119894 119905119894+1] is partitioned into119873119904 sub-subintervals as(119905119894 119905119894+1] = ⋃
119873119904minus1
119895=0(119905119894119895 119905119894119895+1
] where 1199051198940 = 119905119894 and 119905119894119873119904 = 119905
119894+1and denote the sub-subinterval length by ℎ119904 = (119905
119894+1minus 119905119894)119873119904
Denote the value of a variable V on the 119905119894 time point by V119894 andthe one on 119905119894119895 by V119894119895
For the pressure equation the saturations take the valuesof previous time steps and the capillary potentialΦ119888 on eachcell are explicitly calculated by using the cell saturations fromthe previous time step and the capillary pressure functionsThe variables 120582119908 120582119899 and 120582119905 in the pressure equation arealso explicitly calculated by using the cell saturations from
4 Advances in Mathematical Physics
the previous time step From this we obtain the pressureequation in the matrix domain
minusnabla sdot 120582119905 (119878119894
119908119898)K119898nablaΦ
119894+1
119908119898minus nabla sdot 120582119899 (119878
119894
119908119898)K119898nablaΦ
119894
119888119898= 119902119894+1
119905119898
(23)
where the superscript 119894 represents the time step Equation (17)is subject to the matrix-fracture interface condition
Φ119894+1
119908119898= Φ119894+1
119908119891 Φ119894
119888119898= Φ119894
119888119891 on 120597Ω119898 cap Ω119891 (24)
It is similar to express the form in the fracture (referred to bythe subscript 119891) as
minus nabla sdot 120582119905 (119878119894
119908119891)K119891nablaΦ
119894+1
119908119891
minus nabla sdot 120582119899 (119878119894
119908119891)K119891nablaΦ
119894
119888119891= 119902119894+1
119905119891+ 119876119894+1
119905119891
(25)
Once the pressures Φ119894+1119908119898
and Φ119894+1119908119891
are computed thevelocities can be evaluated as
u119894+1119886= minus120582119894
119905KnablaΦ119894+1119908 (26)
As previously mentioned the lower limit 119886 of integralin (1) is a function with time This function can be cho-sen according to practical problems For two-phase flowin porous media the pressure changes less rapidly thanthe saturation with the time [18] and hence a large timestep is taken for the pressure We can also see that thepressures are determined by saturations but have not anyhistorical memory As a result the memory of saturationscan be restricted within each time step for the pressures Fordescribing this memory property we define
119886 (119905) = 119905119894 119905 isin (119905
119894 119905119894+1] (27)
Thus the Caputo fractional derivative of saturation 119878 isdefined by
119863120574
119905119894 119905119878 (119905) =
1
Γ (1 minus 120574)int
119905
119905119894(119905 minus 120591)
minus120574 120597119878
120597119905(120591) 119889120591
119905 isin (119905119894 119905119894+1]
(28)
We now introduce the explicit time discretization scheme forapproximating119863120574
119886119905119878 at 119905119894119895+1 as
119863120574
119905119894 119905119894119895+1119878119894119895+1
=
1198870 (119878119894119895+1
minus 119878119894119895) 119895 = 0
1198870 (119878119894119895+1
minus 119878119894119895)
+1198870
119895
sum
119896=1
119887119895minus119896 (119878119894119895minus119896+1
minus 119878119894119895minus119896
) 119895 = 1 119873119904
(29)
where
1198870 =ℎminus120574
119904
Γ (2 minus 120574)
119887119896 = (119896 + 1)1minus120574
minus 1198961minus120574 119896 = 1 119873119904 minus 1
(30)
The explicit scheme is employed for the saturation equa-tion both in the matrix domain
120601119898119863120574119898
119905119894 119905119894119895+1119878119894119895+1
119908119898+ nabla sdot (119891
119894119895
119908119898u119894+1119886119898) = 119902119894119895+1
119908119898 (31)
and in the fracture network
120601119891119863120574119891
119905119894 119905119894119895+1119878119894119895+1
119908119891+ nabla sdot (119891
119894119895
119908119891u119894+1119886119891) = 119902119894119895+1
119908119891+ 119876119894119895
119908119891 (32)
For spatial discretization schemes the cell-centered finitedifference method is used for the pressure equation whilethe upwind finite volume method is employed for the satu-ration equation For the detailed descriptions about spatialdiscretization schemes we refer to [23]
4 Numerical Tests
In this section two numerical examples are provided todemonstrate the proposed numerical model for two-phaseflow with fractional time derivatives
In all tests the absolute permeability is a diagonal tensorand the porous media are isotropic We use the followingcapillary pressure function [24]
119901119888 (119878119908) = minus119861119888 log (119878119908) (33)
where 119861119888 is a positive parameter related to the absolutepermeability The relative permeabilities of two phases arecomputed by
119896119903119908 = 1198783
119908 119896119903119899 = (1 minus 119878119908)
3 (34)
We consider a horizontal porous medium of 20m times
15m times 1m with multiple interconnected fractures [23]which is shown in Figure 1 The width of fractures is 001mThe porosities of matrix and fracture media are 015 and 1respectively The permeabilities in the matrix blocks and thefractures are 50md and 105 md respectively The viscositiesof the water and oil are all equal to 1 cP The injection rate is02 PVyear
Because the medium is horizontal it is reasonable toneglect the effect of gravity We inject the water at the left endof the medium whose void is initially fully saturated with oilto produce the oil at the right-hand side There is no otherinjection and no extraction to the interior of the domainThefluxes towards outsides of the other boundaries vanish
Figure 2 shows the effects of fractional time generaliza-tion on the average water saturations at different points ofPVI In the legend of Figure 2 the case [120574119891 120574119898] represents theorders of the fractional time derivative of water saturation infracture and matrix regions From Figure 2 we observe theslower temporal decay when compared to the ordinary case
Figures 3 4 5 6 7 8 9 10 and 11 show the watersaturation contours at different time with three pairs of thefractional time derivative From these figures we can seethe presence of a time delay effect when the fractional orderbecomes less than one This indicates that the porosity andpermeability change because of the injected water wetting themedia which makes the fluid flow slowly
Advances in Mathematical Physics 5
Fracture
Length (m)
Wid
th (m
)
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 1 Distribution of fractures
0 05 1 15 2 25PVI
Aver
age s
atur
atio
ns
0
01
02
03
04
05
06
07
08
09
[09 095]
[085 09]
[1 1]
Figure 2 Water saturations with different fractional time deriva-tives
02
02
02
02
02
02
02
04
04
04
04
04
04
04
0606
060606
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 3 Water saturation contour at 125 years with 120574119891= 1 and
120574119898= 1
0606
06
07
07 07
07
07
07
0707
07
07
07
07
07
08
0808
080808
08
0808
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 4 Water saturation contour at 5 years with 120574119891= 1 and 120574
119898=
1
07
08 08
0808
08
08
08
08
08
0808
08
08
08
08
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 5 Water saturation contour at 10 years with 120574119891= 1 and 120574
119898=
1
02
02
02
02
04
04
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
04
04
06
06
Figure 6 Water saturation contour at 125 years with 120574119891= 09 and
120574119898= 095
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
between matrixes and fractures the proposed formulationuses the different fractional time derivatives for the flowin fracture and matrix regions to represent their differentmemory properties
Many numerical methods have been developed in theliterature for example [15ndash17] In order to simulate thefractional two-phase flow we propose a two-level timediscrete method based on the physical property that thepressure varies less rapidly than the saturation [18] thatis a large time step is used for the pressure along with asmall time step size for the saturation The local fractionaldifferential equations have been widely studied in [19] Inour method the memory of saturations is restricted withineach large time step since the pressures are determined bysaturationswithout historicalmemoryTheproposedmethodis also viewed as the fractional generalization of the classicalIMplicit Pressure Explicit Saturation (IMPES) method [2021] which is a popular time-stepping approach employedin multiphase flow simulation Like IMPES we split thecoupled system into one pressure equation and one saturationequation based on the property of multiphysics processesof two-phase flow and treat the saturation and capillarypressure in the pressure equation explicitly to eliminate itsnonlinearity Each large time step is further divided into afew substeps in which the saturation is updated by an explicitfractional time scheme The cell-centered finite differencemethod [22] is employed for spatial discretization Finallynumerical results are given to demonstrate the validity of theproposed numerical model
2 Fractional Model of Two-PhaseIncompressible Flow
21 Fractional Model The Caputo fractional derivative isgiven by the convolution of a power law kernel and the ordi-nary derivative of the function So it is a useful instrument toconsider a power law frequency variability of the coefficientsby a simple convolution We now introduce the definition ofthe Caputo fractional derivative [17] as
119863120574
119886119905119910 (119905) =
1
Γ (1 minus 120574)int
119905
119886
(119905 minus 120591)minus1205741199101015840(120591) 119889120591 (1)
where 119910(119905) is a function and 0 lt 120574 lt 1 Here we use themodified Caputo fractional derivative that is the lower limitof integral in (1) is taken to be a function of 119905 instead of theconstant as usual that is 119886 = 119886(119905) The fractional derivativecan be used to describe the complex problems that involvememory in time because of the nonlocal property From this119886(119905) indicates the memory range at time 119905 and this memoryproperty varies with time
Using the Caputo fractional derivative we introducea memory formalism for two-phase incompressible andimmiscible fluid flow in porous media Denote the wettingphase by a subscript119908 and the nonwetting phase by 119899 Let 119878120572be the saturation of phase 120572 The two-phase saturations aresubject to the following constraint
119878119908 + 119878119899 = 1 (2)
For flow in porous media the velocity u120572 of each phase 120572is described by Darcyrsquos law as
u120572 = minus119896119903120572
120583120572
K (nabla119901120572 + 120588120572119892nabla119911) 120572 = 119908 119899 (3)
where K is the absolute permeability tensor in the porousmedium 119892 is the gravity acceleration 119911 is the depth and119896119903120572 120583120572 119901120572 and 120588120572 are the relative permeability viscositypressure and density of each phase respectively
Themass conservation equation of each phase is given by
120601119863120574
119886119905119878120572 + nabla sdot u120572 = 119902120572 120572 = 119908 119899 (4)
where 120601 is the porosity of the medium and 119902120572 is the externalmass flow rate
The difference between the nonwetting phase andwettingphase pressures is described by the capillary pressure
119901119888 (119878119908) = 119901119899 minus 119901119908 (5)
Denote 120582120572 = 119896119903120572120583120572 and Φ120572 = 119901120572 + 120588119892119911 Substituting (3)into (4) we obtain
120601119863120574
119886119905119878120572 + nabla sdot 120582120572KnablaΦ120572 = 119902120572 120572 = 119908 119899 (6)
Summing the forms (6) of two phases and taking into account(2) we can reach
minusnabla sdot 120582119905KnablaΦ119908 minus nabla sdot 120582119899KnablaΦ119888 = 119902119905 (7)
where 120582119905 = 120582119908 + 120582119899Φ119888 = 119901119888 + (120588119899 minus 120588119908)119892119911 and 119902119905 = 119902119908 + 119902119899Furthermore define u119886 = minus120582119905KnablaΦ119908 and 119891119908 = 120582119908120582119905 Wethen have
u119908 = 119891119908u119886 (8)
Thus the equation for the wetting-phase saturation becomes
120601119863120574
119886119905119878119908 + nabla sdot (119891119908u119886) = 119902119908 (9)
We note that the above two-phase flow formulation is afractional extension of the formulation [4] We consider theclassical mass conservative formulations of incompressibletwo-phase flow
120597120601119878120572
120597119905+ nabla sdot u120572 = 119902120572 120572 = 119908 119899 (10)
The porosity often depends on the fluid pressure and satu-ration In some problems of incompressible two-phase flowpetroleum reservoir engineering for example the porosityvaries small with respect to the pressure However theinjection fluids can change the porosity to a great extentIn the displacement process of oil by water the injectedwater will make the media wet and then the porosity willbe enlarged Summing the mass conservative formulations oftwo phases we obtain
120597120601
120597119905minus nabla sdot 120582119905KnablaΦ119908 minus nabla sdot 120582119899KnablaΦ119888 = 119902119905 (11)
Advances in Mathematical Physics 3
Since the porosity varies small with the pressure we assume120597120601120597119905 ≃ 0 in (11) and as a result a pressure equationis obtained with the same form to (9) The wetting-phasesaturation equation can be expressed as
120597120601119878119908
120597119905+ nabla sdot (119891119908u119886) = 119902119908 (12)
Because of porosity depending on 119878119908 we get by the chain rulethat
120597120601119878119908
120597119905= Υ (119878119908)
120597119878119908
120597119905 (13)
where Υ(119878119908) = 120601 + 1198781199081206011015840(119878119908) In general Υ(119878119908) can be
described by the power law With this argument we candeduce the fractional generalization of the classical two-phase model This clearly explains the physical reason to usethe fractional model
Finally we complete our model by the boundary andinitial conditions We divide the boundary 120597Ω of the com-putational domain Ω into the two nonoverlapping parts theDirichelt part Γ119863 and Newmann part Γ119873 where 120597Ω = Γ
119863cup
Γ119873 The pressure equation (6) is subject to the following
boundary conditions
119901119908 (or 119901119899) = 119901119863 on Γ119863
u120572 sdot n = u119873120572
on Γ119873(14)
where 119901119863 is the pressure on Γ119863 n is the outward unit normalvector to 120597Ω and u119873 is the imposed inflow rate on Γ119873 Theboundary conditions for the saturations are given by
119878119908 = 119878119873 on Γ119873 (15)
The initial saturation of the wetting phase is given by
119878119908 = 1198780
119908in Ω (16)
22 Discrete-Fracture Model with Fractional Time Deriva-tives Here the discrete-fracture model [4] is extended tothe case with the fractional time derivative The discrete-fracture model treats the matrix and fracture gridcells bydifferent geometrical dimensions that is if the domain is 119899-dimensional the matrix regions are n-dimensional but thefractures are simplified as the matrix gridcell interfaces thatare (119899 minus 1)-dimensional This treatment removes the length-scale contrast resulting from the explicit representation ofthe fracture aperture as in the single-porosity model so it iscapable to considerably improve the computational efficiencyand is convenient in practical implementation
We now decompose the entire domain into two parts thematrixΩ119898 and fractureΩ119891 The fractures are surrounded bythe matrix blocks We use the subscript 119898 to represent thematrix and the subscript 119891 to represent the fracture systemThe pressure in the matrix domain is determined by
minusnabla sdot 120582119905119898K119898nablaΦ119908119898 minus nabla sdot 120582119899119898K119898nablaΦ119888119898 = 119902119905119898 (17)
which is subject to the matrix-fracture interface condition
Φ119908119898 = Φ119908119891
Φ119888119898 = Φ119888119891 on 120597Ω119898 cap Ω119891(18)
In the fracture system we denote the fracture width by 120576and assume that the potentials are constant along the fracturewidth and then obtain the pressure equation in the fractureas
minusnabla sdot 120582119905119891K119891nablaΦ119908119891 minus nabla sdot 120582119899119891K119891nablaΦ119888119891 = 119902119905119891 + 119876119905119891 (19)
where 119876119905119891 is the mass transfer across the matrix-fractureinterfaces The above formulations are similar to the classicalmodel
As stated previously the fractional property representsthe variability of porosity and permeability which have aremarkable contrast between matrixes and fractures As aresult different fractional time derivatives should be usedfor the flow in the matrix regions and the fracture systemBy using the fractional time derivative we can express thesaturation equation in the matrix regions as
120601119898119863120574119898
119886119905119878119908119898 + nabla sdot (119891119908119898u119886119898) = 119902119908119898 (20)
along with the interface condition
119878119908119898 = 119878119908119891 on 120597Ω119898 cap Ω119891 (21)
Similarly the saturation equation in the fracture system isgiven by
120601119891119863120574119891
119886119905119878119908119891 + nabla sdot (119891119908119891u119886119891) = 119902119908119891 + 119876119908119891 (22)
where 119876119908119891 represents the mass transfer across the matrix-fracture interfaces
3 Numerical Methods
In this section we will present the numerical methods forthe fractional model of two-phase incompressible flow In thefollowing we focus on the time discretization schemes
We firstly divide the total time interval [0 119879] into 119873119901equal time steps as 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119873119901 = 119879 and denote thetime step length ℎ119901 = 119879119873119901This time division is used for thepressures Since the saturation varies more rapidly than thepressure we use a smaller time step size for saturation Eachsubinterval (119905119894 119905119894+1] is partitioned into119873119904 sub-subintervals as(119905119894 119905119894+1] = ⋃
119873119904minus1
119895=0(119905119894119895 119905119894119895+1
] where 1199051198940 = 119905119894 and 119905119894119873119904 = 119905
119894+1and denote the sub-subinterval length by ℎ119904 = (119905
119894+1minus 119905119894)119873119904
Denote the value of a variable V on the 119905119894 time point by V119894 andthe one on 119905119894119895 by V119894119895
For the pressure equation the saturations take the valuesof previous time steps and the capillary potentialΦ119888 on eachcell are explicitly calculated by using the cell saturations fromthe previous time step and the capillary pressure functionsThe variables 120582119908 120582119899 and 120582119905 in the pressure equation arealso explicitly calculated by using the cell saturations from
4 Advances in Mathematical Physics
the previous time step From this we obtain the pressureequation in the matrix domain
minusnabla sdot 120582119905 (119878119894
119908119898)K119898nablaΦ
119894+1
119908119898minus nabla sdot 120582119899 (119878
119894
119908119898)K119898nablaΦ
119894
119888119898= 119902119894+1
119905119898
(23)
where the superscript 119894 represents the time step Equation (17)is subject to the matrix-fracture interface condition
Φ119894+1
119908119898= Φ119894+1
119908119891 Φ119894
119888119898= Φ119894
119888119891 on 120597Ω119898 cap Ω119891 (24)
It is similar to express the form in the fracture (referred to bythe subscript 119891) as
minus nabla sdot 120582119905 (119878119894
119908119891)K119891nablaΦ
119894+1
119908119891
minus nabla sdot 120582119899 (119878119894
119908119891)K119891nablaΦ
119894
119888119891= 119902119894+1
119905119891+ 119876119894+1
119905119891
(25)
Once the pressures Φ119894+1119908119898
and Φ119894+1119908119891
are computed thevelocities can be evaluated as
u119894+1119886= minus120582119894
119905KnablaΦ119894+1119908 (26)
As previously mentioned the lower limit 119886 of integralin (1) is a function with time This function can be cho-sen according to practical problems For two-phase flowin porous media the pressure changes less rapidly thanthe saturation with the time [18] and hence a large timestep is taken for the pressure We can also see that thepressures are determined by saturations but have not anyhistorical memory As a result the memory of saturationscan be restricted within each time step for the pressures Fordescribing this memory property we define
119886 (119905) = 119905119894 119905 isin (119905
119894 119905119894+1] (27)
Thus the Caputo fractional derivative of saturation 119878 isdefined by
119863120574
119905119894 119905119878 (119905) =
1
Γ (1 minus 120574)int
119905
119905119894(119905 minus 120591)
minus120574 120597119878
120597119905(120591) 119889120591
119905 isin (119905119894 119905119894+1]
(28)
We now introduce the explicit time discretization scheme forapproximating119863120574
119886119905119878 at 119905119894119895+1 as
119863120574
119905119894 119905119894119895+1119878119894119895+1
=
1198870 (119878119894119895+1
minus 119878119894119895) 119895 = 0
1198870 (119878119894119895+1
minus 119878119894119895)
+1198870
119895
sum
119896=1
119887119895minus119896 (119878119894119895minus119896+1
minus 119878119894119895minus119896
) 119895 = 1 119873119904
(29)
where
1198870 =ℎminus120574
119904
Γ (2 minus 120574)
119887119896 = (119896 + 1)1minus120574
minus 1198961minus120574 119896 = 1 119873119904 minus 1
(30)
The explicit scheme is employed for the saturation equa-tion both in the matrix domain
120601119898119863120574119898
119905119894 119905119894119895+1119878119894119895+1
119908119898+ nabla sdot (119891
119894119895
119908119898u119894+1119886119898) = 119902119894119895+1
119908119898 (31)
and in the fracture network
120601119891119863120574119891
119905119894 119905119894119895+1119878119894119895+1
119908119891+ nabla sdot (119891
119894119895
119908119891u119894+1119886119891) = 119902119894119895+1
119908119891+ 119876119894119895
119908119891 (32)
For spatial discretization schemes the cell-centered finitedifference method is used for the pressure equation whilethe upwind finite volume method is employed for the satu-ration equation For the detailed descriptions about spatialdiscretization schemes we refer to [23]
4 Numerical Tests
In this section two numerical examples are provided todemonstrate the proposed numerical model for two-phaseflow with fractional time derivatives
In all tests the absolute permeability is a diagonal tensorand the porous media are isotropic We use the followingcapillary pressure function [24]
119901119888 (119878119908) = minus119861119888 log (119878119908) (33)
where 119861119888 is a positive parameter related to the absolutepermeability The relative permeabilities of two phases arecomputed by
119896119903119908 = 1198783
119908 119896119903119899 = (1 minus 119878119908)
3 (34)
We consider a horizontal porous medium of 20m times
15m times 1m with multiple interconnected fractures [23]which is shown in Figure 1 The width of fractures is 001mThe porosities of matrix and fracture media are 015 and 1respectively The permeabilities in the matrix blocks and thefractures are 50md and 105 md respectively The viscositiesof the water and oil are all equal to 1 cP The injection rate is02 PVyear
Because the medium is horizontal it is reasonable toneglect the effect of gravity We inject the water at the left endof the medium whose void is initially fully saturated with oilto produce the oil at the right-hand side There is no otherinjection and no extraction to the interior of the domainThefluxes towards outsides of the other boundaries vanish
Figure 2 shows the effects of fractional time generaliza-tion on the average water saturations at different points ofPVI In the legend of Figure 2 the case [120574119891 120574119898] represents theorders of the fractional time derivative of water saturation infracture and matrix regions From Figure 2 we observe theslower temporal decay when compared to the ordinary case
Figures 3 4 5 6 7 8 9 10 and 11 show the watersaturation contours at different time with three pairs of thefractional time derivative From these figures we can seethe presence of a time delay effect when the fractional orderbecomes less than one This indicates that the porosity andpermeability change because of the injected water wetting themedia which makes the fluid flow slowly
Advances in Mathematical Physics 5
Fracture
Length (m)
Wid
th (m
)
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 1 Distribution of fractures
0 05 1 15 2 25PVI
Aver
age s
atur
atio
ns
0
01
02
03
04
05
06
07
08
09
[09 095]
[085 09]
[1 1]
Figure 2 Water saturations with different fractional time deriva-tives
02
02
02
02
02
02
02
04
04
04
04
04
04
04
0606
060606
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 3 Water saturation contour at 125 years with 120574119891= 1 and
120574119898= 1
0606
06
07
07 07
07
07
07
0707
07
07
07
07
07
08
0808
080808
08
0808
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 4 Water saturation contour at 5 years with 120574119891= 1 and 120574
119898=
1
07
08 08
0808
08
08
08
08
08
0808
08
08
08
08
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 5 Water saturation contour at 10 years with 120574119891= 1 and 120574
119898=
1
02
02
02
02
04
04
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
04
04
06
06
Figure 6 Water saturation contour at 125 years with 120574119891= 09 and
120574119898= 095
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
Since the porosity varies small with the pressure we assume120597120601120597119905 ≃ 0 in (11) and as a result a pressure equationis obtained with the same form to (9) The wetting-phasesaturation equation can be expressed as
120597120601119878119908
120597119905+ nabla sdot (119891119908u119886) = 119902119908 (12)
Because of porosity depending on 119878119908 we get by the chain rulethat
120597120601119878119908
120597119905= Υ (119878119908)
120597119878119908
120597119905 (13)
where Υ(119878119908) = 120601 + 1198781199081206011015840(119878119908) In general Υ(119878119908) can be
described by the power law With this argument we candeduce the fractional generalization of the classical two-phase model This clearly explains the physical reason to usethe fractional model
Finally we complete our model by the boundary andinitial conditions We divide the boundary 120597Ω of the com-putational domain Ω into the two nonoverlapping parts theDirichelt part Γ119863 and Newmann part Γ119873 where 120597Ω = Γ
119863cup
Γ119873 The pressure equation (6) is subject to the following
boundary conditions
119901119908 (or 119901119899) = 119901119863 on Γ119863
u120572 sdot n = u119873120572
on Γ119873(14)
where 119901119863 is the pressure on Γ119863 n is the outward unit normalvector to 120597Ω and u119873 is the imposed inflow rate on Γ119873 Theboundary conditions for the saturations are given by
119878119908 = 119878119873 on Γ119873 (15)
The initial saturation of the wetting phase is given by
119878119908 = 1198780
119908in Ω (16)
22 Discrete-Fracture Model with Fractional Time Deriva-tives Here the discrete-fracture model [4] is extended tothe case with the fractional time derivative The discrete-fracture model treats the matrix and fracture gridcells bydifferent geometrical dimensions that is if the domain is 119899-dimensional the matrix regions are n-dimensional but thefractures are simplified as the matrix gridcell interfaces thatare (119899 minus 1)-dimensional This treatment removes the length-scale contrast resulting from the explicit representation ofthe fracture aperture as in the single-porosity model so it iscapable to considerably improve the computational efficiencyand is convenient in practical implementation
We now decompose the entire domain into two parts thematrixΩ119898 and fractureΩ119891 The fractures are surrounded bythe matrix blocks We use the subscript 119898 to represent thematrix and the subscript 119891 to represent the fracture systemThe pressure in the matrix domain is determined by
minusnabla sdot 120582119905119898K119898nablaΦ119908119898 minus nabla sdot 120582119899119898K119898nablaΦ119888119898 = 119902119905119898 (17)
which is subject to the matrix-fracture interface condition
Φ119908119898 = Φ119908119891
Φ119888119898 = Φ119888119891 on 120597Ω119898 cap Ω119891(18)
In the fracture system we denote the fracture width by 120576and assume that the potentials are constant along the fracturewidth and then obtain the pressure equation in the fractureas
minusnabla sdot 120582119905119891K119891nablaΦ119908119891 minus nabla sdot 120582119899119891K119891nablaΦ119888119891 = 119902119905119891 + 119876119905119891 (19)
where 119876119905119891 is the mass transfer across the matrix-fractureinterfaces The above formulations are similar to the classicalmodel
As stated previously the fractional property representsthe variability of porosity and permeability which have aremarkable contrast between matrixes and fractures As aresult different fractional time derivatives should be usedfor the flow in the matrix regions and the fracture systemBy using the fractional time derivative we can express thesaturation equation in the matrix regions as
120601119898119863120574119898
119886119905119878119908119898 + nabla sdot (119891119908119898u119886119898) = 119902119908119898 (20)
along with the interface condition
119878119908119898 = 119878119908119891 on 120597Ω119898 cap Ω119891 (21)
Similarly the saturation equation in the fracture system isgiven by
120601119891119863120574119891
119886119905119878119908119891 + nabla sdot (119891119908119891u119886119891) = 119902119908119891 + 119876119908119891 (22)
where 119876119908119891 represents the mass transfer across the matrix-fracture interfaces
3 Numerical Methods
In this section we will present the numerical methods forthe fractional model of two-phase incompressible flow In thefollowing we focus on the time discretization schemes
We firstly divide the total time interval [0 119879] into 119873119901equal time steps as 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119873119901 = 119879 and denote thetime step length ℎ119901 = 119879119873119901This time division is used for thepressures Since the saturation varies more rapidly than thepressure we use a smaller time step size for saturation Eachsubinterval (119905119894 119905119894+1] is partitioned into119873119904 sub-subintervals as(119905119894 119905119894+1] = ⋃
119873119904minus1
119895=0(119905119894119895 119905119894119895+1
] where 1199051198940 = 119905119894 and 119905119894119873119904 = 119905
119894+1and denote the sub-subinterval length by ℎ119904 = (119905
119894+1minus 119905119894)119873119904
Denote the value of a variable V on the 119905119894 time point by V119894 andthe one on 119905119894119895 by V119894119895
For the pressure equation the saturations take the valuesof previous time steps and the capillary potentialΦ119888 on eachcell are explicitly calculated by using the cell saturations fromthe previous time step and the capillary pressure functionsThe variables 120582119908 120582119899 and 120582119905 in the pressure equation arealso explicitly calculated by using the cell saturations from
4 Advances in Mathematical Physics
the previous time step From this we obtain the pressureequation in the matrix domain
minusnabla sdot 120582119905 (119878119894
119908119898)K119898nablaΦ
119894+1
119908119898minus nabla sdot 120582119899 (119878
119894
119908119898)K119898nablaΦ
119894
119888119898= 119902119894+1
119905119898
(23)
where the superscript 119894 represents the time step Equation (17)is subject to the matrix-fracture interface condition
Φ119894+1
119908119898= Φ119894+1
119908119891 Φ119894
119888119898= Φ119894
119888119891 on 120597Ω119898 cap Ω119891 (24)
It is similar to express the form in the fracture (referred to bythe subscript 119891) as
minus nabla sdot 120582119905 (119878119894
119908119891)K119891nablaΦ
119894+1
119908119891
minus nabla sdot 120582119899 (119878119894
119908119891)K119891nablaΦ
119894
119888119891= 119902119894+1
119905119891+ 119876119894+1
119905119891
(25)
Once the pressures Φ119894+1119908119898
and Φ119894+1119908119891
are computed thevelocities can be evaluated as
u119894+1119886= minus120582119894
119905KnablaΦ119894+1119908 (26)
As previously mentioned the lower limit 119886 of integralin (1) is a function with time This function can be cho-sen according to practical problems For two-phase flowin porous media the pressure changes less rapidly thanthe saturation with the time [18] and hence a large timestep is taken for the pressure We can also see that thepressures are determined by saturations but have not anyhistorical memory As a result the memory of saturationscan be restricted within each time step for the pressures Fordescribing this memory property we define
119886 (119905) = 119905119894 119905 isin (119905
119894 119905119894+1] (27)
Thus the Caputo fractional derivative of saturation 119878 isdefined by
119863120574
119905119894 119905119878 (119905) =
1
Γ (1 minus 120574)int
119905
119905119894(119905 minus 120591)
minus120574 120597119878
120597119905(120591) 119889120591
119905 isin (119905119894 119905119894+1]
(28)
We now introduce the explicit time discretization scheme forapproximating119863120574
119886119905119878 at 119905119894119895+1 as
119863120574
119905119894 119905119894119895+1119878119894119895+1
=
1198870 (119878119894119895+1
minus 119878119894119895) 119895 = 0
1198870 (119878119894119895+1
minus 119878119894119895)
+1198870
119895
sum
119896=1
119887119895minus119896 (119878119894119895minus119896+1
minus 119878119894119895minus119896
) 119895 = 1 119873119904
(29)
where
1198870 =ℎminus120574
119904
Γ (2 minus 120574)
119887119896 = (119896 + 1)1minus120574
minus 1198961minus120574 119896 = 1 119873119904 minus 1
(30)
The explicit scheme is employed for the saturation equa-tion both in the matrix domain
120601119898119863120574119898
119905119894 119905119894119895+1119878119894119895+1
119908119898+ nabla sdot (119891
119894119895
119908119898u119894+1119886119898) = 119902119894119895+1
119908119898 (31)
and in the fracture network
120601119891119863120574119891
119905119894 119905119894119895+1119878119894119895+1
119908119891+ nabla sdot (119891
119894119895
119908119891u119894+1119886119891) = 119902119894119895+1
119908119891+ 119876119894119895
119908119891 (32)
For spatial discretization schemes the cell-centered finitedifference method is used for the pressure equation whilethe upwind finite volume method is employed for the satu-ration equation For the detailed descriptions about spatialdiscretization schemes we refer to [23]
4 Numerical Tests
In this section two numerical examples are provided todemonstrate the proposed numerical model for two-phaseflow with fractional time derivatives
In all tests the absolute permeability is a diagonal tensorand the porous media are isotropic We use the followingcapillary pressure function [24]
119901119888 (119878119908) = minus119861119888 log (119878119908) (33)
where 119861119888 is a positive parameter related to the absolutepermeability The relative permeabilities of two phases arecomputed by
119896119903119908 = 1198783
119908 119896119903119899 = (1 minus 119878119908)
3 (34)
We consider a horizontal porous medium of 20m times
15m times 1m with multiple interconnected fractures [23]which is shown in Figure 1 The width of fractures is 001mThe porosities of matrix and fracture media are 015 and 1respectively The permeabilities in the matrix blocks and thefractures are 50md and 105 md respectively The viscositiesof the water and oil are all equal to 1 cP The injection rate is02 PVyear
Because the medium is horizontal it is reasonable toneglect the effect of gravity We inject the water at the left endof the medium whose void is initially fully saturated with oilto produce the oil at the right-hand side There is no otherinjection and no extraction to the interior of the domainThefluxes towards outsides of the other boundaries vanish
Figure 2 shows the effects of fractional time generaliza-tion on the average water saturations at different points ofPVI In the legend of Figure 2 the case [120574119891 120574119898] represents theorders of the fractional time derivative of water saturation infracture and matrix regions From Figure 2 we observe theslower temporal decay when compared to the ordinary case
Figures 3 4 5 6 7 8 9 10 and 11 show the watersaturation contours at different time with three pairs of thefractional time derivative From these figures we can seethe presence of a time delay effect when the fractional orderbecomes less than one This indicates that the porosity andpermeability change because of the injected water wetting themedia which makes the fluid flow slowly
Advances in Mathematical Physics 5
Fracture
Length (m)
Wid
th (m
)
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 1 Distribution of fractures
0 05 1 15 2 25PVI
Aver
age s
atur
atio
ns
0
01
02
03
04
05
06
07
08
09
[09 095]
[085 09]
[1 1]
Figure 2 Water saturations with different fractional time deriva-tives
02
02
02
02
02
02
02
04
04
04
04
04
04
04
0606
060606
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 3 Water saturation contour at 125 years with 120574119891= 1 and
120574119898= 1
0606
06
07
07 07
07
07
07
0707
07
07
07
07
07
08
0808
080808
08
0808
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 4 Water saturation contour at 5 years with 120574119891= 1 and 120574
119898=
1
07
08 08
0808
08
08
08
08
08
0808
08
08
08
08
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 5 Water saturation contour at 10 years with 120574119891= 1 and 120574
119898=
1
02
02
02
02
04
04
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
04
04
06
06
Figure 6 Water saturation contour at 125 years with 120574119891= 09 and
120574119898= 095
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
the previous time step From this we obtain the pressureequation in the matrix domain
minusnabla sdot 120582119905 (119878119894
119908119898)K119898nablaΦ
119894+1
119908119898minus nabla sdot 120582119899 (119878
119894
119908119898)K119898nablaΦ
119894
119888119898= 119902119894+1
119905119898
(23)
where the superscript 119894 represents the time step Equation (17)is subject to the matrix-fracture interface condition
Φ119894+1
119908119898= Φ119894+1
119908119891 Φ119894
119888119898= Φ119894
119888119891 on 120597Ω119898 cap Ω119891 (24)
It is similar to express the form in the fracture (referred to bythe subscript 119891) as
minus nabla sdot 120582119905 (119878119894
119908119891)K119891nablaΦ
119894+1
119908119891
minus nabla sdot 120582119899 (119878119894
119908119891)K119891nablaΦ
119894
119888119891= 119902119894+1
119905119891+ 119876119894+1
119905119891
(25)
Once the pressures Φ119894+1119908119898
and Φ119894+1119908119891
are computed thevelocities can be evaluated as
u119894+1119886= minus120582119894
119905KnablaΦ119894+1119908 (26)
As previously mentioned the lower limit 119886 of integralin (1) is a function with time This function can be cho-sen according to practical problems For two-phase flowin porous media the pressure changes less rapidly thanthe saturation with the time [18] and hence a large timestep is taken for the pressure We can also see that thepressures are determined by saturations but have not anyhistorical memory As a result the memory of saturationscan be restricted within each time step for the pressures Fordescribing this memory property we define
119886 (119905) = 119905119894 119905 isin (119905
119894 119905119894+1] (27)
Thus the Caputo fractional derivative of saturation 119878 isdefined by
119863120574
119905119894 119905119878 (119905) =
1
Γ (1 minus 120574)int
119905
119905119894(119905 minus 120591)
minus120574 120597119878
120597119905(120591) 119889120591
119905 isin (119905119894 119905119894+1]
(28)
We now introduce the explicit time discretization scheme forapproximating119863120574
119886119905119878 at 119905119894119895+1 as
119863120574
119905119894 119905119894119895+1119878119894119895+1
=
1198870 (119878119894119895+1
minus 119878119894119895) 119895 = 0
1198870 (119878119894119895+1
minus 119878119894119895)
+1198870
119895
sum
119896=1
119887119895minus119896 (119878119894119895minus119896+1
minus 119878119894119895minus119896
) 119895 = 1 119873119904
(29)
where
1198870 =ℎminus120574
119904
Γ (2 minus 120574)
119887119896 = (119896 + 1)1minus120574
minus 1198961minus120574 119896 = 1 119873119904 minus 1
(30)
The explicit scheme is employed for the saturation equa-tion both in the matrix domain
120601119898119863120574119898
119905119894 119905119894119895+1119878119894119895+1
119908119898+ nabla sdot (119891
119894119895
119908119898u119894+1119886119898) = 119902119894119895+1
119908119898 (31)
and in the fracture network
120601119891119863120574119891
119905119894 119905119894119895+1119878119894119895+1
119908119891+ nabla sdot (119891
119894119895
119908119891u119894+1119886119891) = 119902119894119895+1
119908119891+ 119876119894119895
119908119891 (32)
For spatial discretization schemes the cell-centered finitedifference method is used for the pressure equation whilethe upwind finite volume method is employed for the satu-ration equation For the detailed descriptions about spatialdiscretization schemes we refer to [23]
4 Numerical Tests
In this section two numerical examples are provided todemonstrate the proposed numerical model for two-phaseflow with fractional time derivatives
In all tests the absolute permeability is a diagonal tensorand the porous media are isotropic We use the followingcapillary pressure function [24]
119901119888 (119878119908) = minus119861119888 log (119878119908) (33)
where 119861119888 is a positive parameter related to the absolutepermeability The relative permeabilities of two phases arecomputed by
119896119903119908 = 1198783
119908 119896119903119899 = (1 minus 119878119908)
3 (34)
We consider a horizontal porous medium of 20m times
15m times 1m with multiple interconnected fractures [23]which is shown in Figure 1 The width of fractures is 001mThe porosities of matrix and fracture media are 015 and 1respectively The permeabilities in the matrix blocks and thefractures are 50md and 105 md respectively The viscositiesof the water and oil are all equal to 1 cP The injection rate is02 PVyear
Because the medium is horizontal it is reasonable toneglect the effect of gravity We inject the water at the left endof the medium whose void is initially fully saturated with oilto produce the oil at the right-hand side There is no otherinjection and no extraction to the interior of the domainThefluxes towards outsides of the other boundaries vanish
Figure 2 shows the effects of fractional time generaliza-tion on the average water saturations at different points ofPVI In the legend of Figure 2 the case [120574119891 120574119898] represents theorders of the fractional time derivative of water saturation infracture and matrix regions From Figure 2 we observe theslower temporal decay when compared to the ordinary case
Figures 3 4 5 6 7 8 9 10 and 11 show the watersaturation contours at different time with three pairs of thefractional time derivative From these figures we can seethe presence of a time delay effect when the fractional orderbecomes less than one This indicates that the porosity andpermeability change because of the injected water wetting themedia which makes the fluid flow slowly
Advances in Mathematical Physics 5
Fracture
Length (m)
Wid
th (m
)
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 1 Distribution of fractures
0 05 1 15 2 25PVI
Aver
age s
atur
atio
ns
0
01
02
03
04
05
06
07
08
09
[09 095]
[085 09]
[1 1]
Figure 2 Water saturations with different fractional time deriva-tives
02
02
02
02
02
02
02
04
04
04
04
04
04
04
0606
060606
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 3 Water saturation contour at 125 years with 120574119891= 1 and
120574119898= 1
0606
06
07
07 07
07
07
07
0707
07
07
07
07
07
08
0808
080808
08
0808
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 4 Water saturation contour at 5 years with 120574119891= 1 and 120574
119898=
1
07
08 08
0808
08
08
08
08
08
0808
08
08
08
08
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 5 Water saturation contour at 10 years with 120574119891= 1 and 120574
119898=
1
02
02
02
02
04
04
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
04
04
06
06
Figure 6 Water saturation contour at 125 years with 120574119891= 09 and
120574119898= 095
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
Fracture
Length (m)
Wid
th (m
)
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 1 Distribution of fractures
0 05 1 15 2 25PVI
Aver
age s
atur
atio
ns
0
01
02
03
04
05
06
07
08
09
[09 095]
[085 09]
[1 1]
Figure 2 Water saturations with different fractional time deriva-tives
02
02
02
02
02
02
02
04
04
04
04
04
04
04
0606
060606
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 3 Water saturation contour at 125 years with 120574119891= 1 and
120574119898= 1
0606
06
07
07 07
07
07
07
0707
07
07
07
07
07
08
0808
080808
08
0808
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 4 Water saturation contour at 5 years with 120574119891= 1 and 120574
119898=
1
07
08 08
0808
08
08
08
08
08
0808
08
08
08
08
08
08
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 5 Water saturation contour at 10 years with 120574119891= 1 and 120574
119898=
1
02
02
02
02
04
04
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
04
04
06
06
Figure 6 Water saturation contour at 125 years with 120574119891= 09 and
120574119898= 095
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
02
02
02
02
02
04
04
04
060
6
06
06
06
06
06
06
08
08
08
080
8
08
04
04
04
06
06
06
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 7 Water saturation contour at 5 years with 120574119891= 09 and
120574119898= 095
07 0707
07 07
07
07
07
07
07
07
08
0808
08
08
0808
08
08
08
0808
09
09
09
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 8 Water saturation contour at 10 years with 120574119891= 09 and
120574119898= 095
02
02
02
04
04
04
06
06
06
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 9 Water saturation contour at 125 years with 120574119891= 085 and
120574119898= 09
02
02
02
02
02
02
0202
04
04
04
04 04
04
04
04
06
0606
06 06
06
06
06
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 10 Water saturation contour at 5 years with 120574119891= 085 and
120574119898= 09
02
04
04
04
02
04
06
06
06
0606
0606
06
06
08
0808
08
08
08
08
08
5
10
15
00
2 4 6 8 10 12 14 16 18 20
Figure 11 Water saturation contour at 10 years with 120574119891= 085 and
120574119898= 09
Acknowledgments
This work is supported by the Key Program of Shang-hai Municipal Education Commission under Grant (no12ZZ084) and the Key Project of Chinese Ministry of Edu-cation (no 212109)
References
[1] R G Baca R C Arnett and D W Langford ldquoModellingfluid flow in fractured-porous rock masses by finite-elementtechniquesrdquo International Journal for Numerical Methods inFluids vol 4 no 4 pp 337ndash348 1984
[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960
[3] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media SIAM Philadelphia PaUSA 2006
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
[4] H Hoteit and A Firoozabadi ldquoAn efficient numerical model forincompressible two-phase flow in fracturedmediardquoAdvances inWater Resources vol 31 no 6 pp 891ndash905 2008
[5] H Kazemi ldquoPressure transient analysis of naturally fracturedreservoirs with uniform fracture distributionrdquo SPE Journal vol9 no 4 pp 451ndash462 1969
[6] S H Lee C L Jensen and M F Lough ldquoEfficient finite-difference model for flow in a reservoir with multiple length-scale fracturesrdquo SPE Journal vol 5 no 3 pp 268ndash275 2000
[7] J Noorishad and M Mehran ldquoAn upstream finite elementmethod for solution of transient transport equation in fracturedporous mediardquoWater Resources Research vol 18 no 3 pp 588ndash596 1982
[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porous mediardquo SPEJournal vol 25 no 1 pp 14ndash26 1985
[9] S Sun A Firoozabadi and J Kou ldquoNumerical modeling oftwo-phase binary fluid mixing using mixed finite elementsrdquoComputational Geosciences vol 16 no 4 pp 1101ndash1124 2012
[10] L K Thomas T N Dixon and R G Pierson ldquoFracturedreservoir simulationrdquo SPE Journal vol 23 no 1 pp 42ndash54 1983
[11] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 1963
[12] E di Giuseppe M Moroni and M Caputo ldquoFlux in porousmedia with memory models and experimentsrdquo Transport inPorous Media vol 83 no 3 pp 479ndash500 2010
[13] M Caputo ldquoModels of flux in porous media with memoryrdquoWater Resources Research vol 36 no 3 pp 693ndash705 2000
[14] R Garra ldquoFractional-calculus model for temperature andpressure waves in fluid-saturated porous rocksrdquo Physical ReviewE vol 84 no 3 Article ID 036605 6 pages 2011
[15] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6ndash8 pp 743ndash773 2005
[16] H Ding and C Li ldquoMixed spline functionmethod for reaction-subdiffusion equationsrdquo Journal of Computational Physics vol242 pp 103ndash123 2013
[17] C Li A Chen and J Ye ldquoNumerical approaches to fractionalcalculus and fractional ordinary differential equationrdquo Journalof Computational Physics vol 230 no 9 pp 3352ndash3368 2011
[18] Z Chen G Huan and B Li ldquoAn improved IMPES method fortwo-phase flow in porous mediardquo Transport in Porous Mediavol 54 no 3 pp 361ndash376 2004
[19] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013
[20] K H Coats ldquoReservoir simulation state-of-the-artrdquo Journal ofPetroleum Technology vol 34 no 8 pp 1633ndash1642 1982
[21] R G Fagin and C H Stewart Jr ldquoA new approach to the two-dimensional multiphase reservoir simulatorrdquo SPE Journal vol6 no 2 pp 175ndash182 1966
[22] P A Forsyth Jr and P H Sammon ldquoQuadratic convergence forcell-centered gridsrdquo Applied Numerical Mathematics vol 4 no5 pp 377ndash394 1988
[23] J Kou S Sun and B Yu ldquoMultiscale time-splitting strategyfor multiscale multiphysics processes of two-phase flow infractured mediardquo Journal of Applied Mathematics vol 2011Article ID 861905 24 pages 2011
[24] H Hoteit and A Firoozabadi ldquoNumerical modeling of two-phase flow in heterogeneous permeable media with differentcapillarity pressuresrdquo Advances in Water Resources vol 31 no1 pp 56ndash73 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
