Research ArticleAcceleration of Gas Reservoir Simulation Using ProperOrthogonal Decomposition

Yi Wang 12 Bo Yu 3 and Ye Wang4

1National Engineering Laboratory for Pipeline Safety andMOE Key Laboratory of Petroleum Engineering and Beijing Key Laboratoryof Urban Oil and Gas Distribution Technology China University of Petroleum Beijing 102249 China2Key Laboratory of Thermo-Fluid Science and Engineering Xirsquoan Jiaotong University Ministry of Education Xirsquoan 710049 China3School of Mechanical Engineering Beijing Institute of Petrochemical Technology Beijing 102617 China4Key Laboratory of Railway Vehicle Thermal Engineering Lanzhou Jiaotong University Ministry of EducationLanzhou 730070 China

Correspondence should be addressed to Bo Yu yuboboxvip163com

Received 3 July 2017 Revised 26 August 2017 Accepted 17 September 2017 Published 3 January 2018

Academic Editor Jianchao Cai

Copyright copy 2018 YiWang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

High-precision and high-speed reservoir simulation is important in engineering Proper orthogonal decomposition (POD) isintroduced to accelerate the reservoir simulation of gas flow in single-continuum porous media via establishing a reduced-ordermodel by POD combined with Galerkin projection Determination of the optimal mode number in the reduced-order modelis discussed to ensure high-precision reconstruction with large acceleration The typical POD model can achieve high precisionfor both ideal gas and real gas using only 10 POD modes However acceleration of computation can only be achieved for idealgas The obstacle of POD acceleration for real gas is that the computational time is mainly occupied by the equation of state(EOS) An approximation method is proposed to largely promote the computational speed of the POD model for real gas flowwithout decreasing the precision The improved POD model shows much higher acceleration of computation with high precisionfor different reservoirs and different pressures It is confirmed that the acceleration of the real gas reservoir simulation should usethe approximation method instead of the computation of EOS

1 Introduction

Reservoir simulation utilizing mathematical models to pre-dict fluid flow in petroleum reservoirs has been developedsince the 1800s [1] It plays more and more importantroles in the development of oil exploration and production[2] Correspondingly mathematical theories and numeri-cal methods have attracted extensive attention [3ndash10] Thecomputational cost of the oil reservoir simulations is hugebecause a very dense mesh should be used causing long-timeiterations to achieve enough resolution This kind of long-time computation is usually not endurable in engineeringPetroleum engineers attempted several approaches to speedup oil reservoir simulations including the scale-up approach[11ndash15] and parallel computation [16] Proper orthogonaldecomposition (POD) can reduce computational loads effi-ciently via a reduced-order model It has been applied to

heated-crude-oil pipe flow [17 18] and other fields [19ndash21]Ghommem et al [22 23] and Efendiev et al [24] discussedthe POD and dynamic mode decomposition method fortime-dependent incompressible single-phase flow in high-contrast heterogeneous porous media They proved that thePOD model can achieve good precision in an appropriaterangeWith fast progress of the exploration and developmentof gas reservoirs gas reservoir simulation attracted moreand more attention [25ndash28] Additional computation shouldbe made to calculate the compressibility of the gas whichdoes not exist in incompressible oil flowThus the numericalsimulations of gas reservoirs are more complex and time-consuming The study on the acceleration method of gasreservoir simulation by POD is meaningful On the otherhand the compressibility of the gas leads to a POD modelwith stronger nonlinearity than oil flow This situation mightcause the POD modeling process to be more difficult than

HindawiGeofluidsVolume 2018 Article ID 8482352 15 pageshttpsdoiorg10115520188482352

2 Geofluids

the previous POD modeling for incompressible single-phaseflow To the best of the authorsrsquo knowledge there is no reportconcerning the POD model for gas reservoir simulationespecially for real gas Therefore it is worth discussing thePOD modeling method for gas reservoir simulations

This paper is organized as follows We revisit the govern-ing equations of gas reservoir simulation and explain howto apply the POD to the gas reservoir simulation brieflyThen a POD model will be established for gas reservoirsimulation in single-continuum porous media Key effectswill be discussed on the acceleration ability of the PODmodel Finally conclusions and suggestions will be made

2 Numerical Methods

21 Governing Equations of Gas Reservoir Simulation Puregas flow obeys mass conservation law (see (1)) Darcyrsquos law(see (2)) and the gas law (see (3)) as follows

120601120597120588120597119905 = minusnabla sdot (120588u) + 119902 (1)

u = minusk120583nabla119901 (2)

120588 = 119901119882119885119877119879 (3)

where u is the Darcy velocity of gas flow in reservoirs k is thediagonal permeability tensor119901 is the pressure120588 is the densityof gas 120583 is the dynamic viscosity of gas 120601 is the porosity ofporous media 119902 is the injection or production rate119882 is themolecular weight of gas 119877 is the universal gas constant 119879 isthe temperature of gas and 119885 is the compressibility factor ofgas 119885 should be calculated by the equation of state of gas Inthis paper we use the Van der Waals equation

1198853 minus (119861 + 1)1198852 + 119860119885 minus 119860119861 = 0 (4)

where 119860 = 11988611990111987721198792 119861 = 119887119901119877119879 119885 = 119901V119877119879 V is themolar volume of gas and 119886 and 119887 are gas constants changingwith different real gases The compressibility of gas forisothermal flow can be defined as

119888119892 = 1120588 11988912058811988911990110038161003816100381610038161003816100381610038161003816119879 (5)

Using pressure as the primary variable (1) and (5) can berewritten as follows

120601119888119892119901120597119901120597119905 = nabla sdot (k120583119901nabla119901) + 119885119877119879119882 119902 (6)

119888119892 = 1119901 minus 1119885 [ 119861 minus 11988531198852 minus 2 (119861 + 1)119885 + 119860 11988611987721198792+ 119860 + 119885231198852 minus 2 (119861 + 1)119885 + 119860 119887119877119879]

(7)

For the convenience of expression we only use two-dimensional cases in this paper but the extension to three-dimensional cases is straightforwardThus the above govern-ing equations have the following expressions

120601119888119892119901120597119901120597119905 = 120597120597119909 (119896119909119909120583 119901120597119901120597119909) + 120597120597119910 (119896119910119910120583 119901120597119901120597119910) + 119885119877119879119882 119902 (8)

119906 = minus119896119909119909120583 120597119901120597119909 (9)

V = minus119896119910119910120583 120597119901120597119910 (10)

where 119896119909119909 and 119896119910119910 are two components of permeability kin the 119909 and 119910 directions respectively and 119906 and V are twocomponents of Darcy velocity u in the 119909 and 119910 directionsrespectively Gas reservoir simulation can be accomplishedvia numerically solving (8)sim(10) to obtain pressure andvelocity with the calculation of parameters (119885 and 119888119892) using(4) and (7)Here 119888119892 is the implicit function of119901 (shown in (7))so that it can only be explicitly solved Therefore the wholeterm 119888119892119901 is treated explicitly

22 Basic Idea of Proper Orthogonal Decomposition Theabove governing equations clearly show that the core of thewhole computation is the calculation of 119901 Once 119901 is solvedone can directly solve 119906 and V Therefore the accelerationof the computation of 119901 is the most important in the PODmodeling The basic idea of the POD modeling for (8) isbriefly stated as follows

(1) Separate the variable in space and time using thefollowing linear combination

119901 = 119872sum119899=1

119888119899120593119899 (11)

where120593119899 are the PODmodes which are the functions of space(120593119899(119909 119910)) 119888119899 are the temporal coefficients (119888119899(119905)) and 120593119899 arethe identity basis vectors orthogonal with each other

120593119894 sdot 120593119895 = 120575119894119895 (12)

where ldquosdotrdquomeans the inner product of two vectors and 120575119894119895 is theKronecker delta taking the value 1 for 119895 = 119894 and 0 for others

(2) 120593119899 can be calculated from the eigenvalue decomposi-tion of a sample matrix which can be obtained by aligningthe samples in a time series (1199051 1199052 1199053 119905119873)

S = [1198781 1198782 1198783 119878119873] (13)

where 1198781 = [[11990111119901211199011198711

]] 1198782 = [

[11990112119901221199011198712

]] 1198783 = [

[11990113119901231199011198713

]] 119878119873 =

[[11990111198731199012119873119901119871119873

]] 119871 is the number of grid points (including boundary

points) and 119873 is the number of samples A kernel can becalculated as follows

119862119894119895 = 1119873 intΩ119878119894119878119895119889Ω (14)

Geofluids 3

whereΩmeans domain The matrix form of (14) is

C = Δ119909Δ119910119873 S119879S (15)

The matrix C can be decomposed by eigenvalue decom-position to obtain the eigenvalues and eigenvectors

CV = 120582V (16)

whereV is the eigenvectormatrix and 120582 are the eigenvalues ofthe corresponding eigenvectors Along with the descendingorder of 120582 (1205821 gt 1205822 gt 1205823 gt sdot sdot sdot gt 120582119873) the importanceof eigenvectors V1 V2 V3 V119873 in V is also descending Toquantitatively represent the importance of each eigenvectorthe energy 120585119899 of the 119899th eigenvector is introduced

120585119899 = 120582119899sum119873119894=1 120582119894 times 100 (17)

Correspondingly the cumulative energy 120577119899 of the top 119899eigenvectors can be expressed as follows

120577119899 = sum119899119895=1 120582119895sum119873119894=1 120582119894 times 100 (18)

POD modes 120593119899 can be obtained as follows

120593119899 = SVSV (19)

where ldquordquo means the Euclidean distance

(3) Substituting (11) into (8) and projecting the wholeequation onto the Hilbert space spanned by the POD modes120601119898 (119898 = 1 sim 119872) one can finally obtain a linear equationsystem of 119888119899 in the scale of119872 times119872 where119872 is the numberof POD modes in (11) Once the temporal coefficients 119888119899 aresolved the original variable 119901 can be directly reconstructedin (11) instead of the computation of (8) The computationalcost of (11) is in the scale of119872times119872 while the computationalcost of (8) is in the scale of 119871 times 119871 The number of samples isusually quite smaller than the number of grid points (119873 ≪119871) The number of effective POD modes is smaller than thenumber of samples (119872 le 119873) Thus the computational timecan be largely reduced using POD The POD model is atype of reduced-order model The details of this step will beexplained in Section 23

According to the above analyses the core computationof POD is the calculation of temporal coefficients 119888119899 Theequation of 119888119899 for gas reservoir simulation will be establishedin detail in the next section

23 Establishment of the POD Model for Gas ReservoirSimulation Substituted by (11) (8) can be transformed to

120601119888119892119901119872sum119899=1

120593119899 119889119888119899119889119905= 119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]

+ 119885119877119879119882 119902

(20)

Equation (20) is projected onto 120593119898

int1198971199100int1198971199090120593119898120601119888119892119901119872sum

119899=1120593119899 119889119888119899119889119905 119889119909 119889119910 = int1198971199100 int1198971199090 120593119898

119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910 + int1198971199100 int1198971199090 120593119898119885119877119879119882 119902119889119909119889119910 (21)

where 119897119909 and 119897119910 are the lengths of the computational domainin the 119909 and 119910 directions respectively As stated in (11) 119888119899 and 120593119899 are functions of time and space respectivelyThus the left-

hand side of (21) is transformed to

int1198971199100int1198971199090120593119898120601119888119892119901119872sum

119899=1120593119899 119889119888119899119889119905 119889119909 119889119910 = 120601

119872sum119899=1

119889119888119899119889119905 int1198971199100 int1198971199090 119888119892119901120593119898120593119899119889119909 119889119910 (22)

Using integration by part and the property of the PODmodes (see (12)) we can simplify the first term and

the second term of the right-hand side of (21) as fol-lows

int1198971199100int1198971199090120593119898 119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910

= 119872sum119899=1

119872sum119897=1

119888119899119888119897 int1198971199100int1198971199090120593119898 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (

119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910 = int119897119909

0[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

4 Geofluids

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909 + int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897 int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

(23)

int1198971199100int1198971199090120593119898119885119877119879119882 119902119889119909119889119910 = 119877119879119882 int119897119910

0int1198971199090120593119898119885119902119889119909119889119910 (24)

Please note that the projection of the boundary conditionin (23) is dependent on boundary pressures and boundarypressure gradients To update them the pressure field shouldbe firstly reconstructed using (11) in every time step once 119888119899 isupdated by the POD model Therefore the projection of theboundary condition leads to additional computations of thefull-order equations (see (9) (10) and (11)) consumingmuchmore time This is very harmful for the high acceleration ofthe PODmodel To ensure the high-acceleration advantage ofthe PODmodel the computations of (9) (10) and (11) shouldbe avoided in the PODmodelThis can be fulfilled by furtherdecomposing the pressures in (23) The derivations will bemade in the following three steps

(1) If all the boundaries are Dirichlet condition (knownboundary pressure) boundary treatment can be expressed asfollows

(120597119901120597119909)119899119909+1119895 =119901119899119909+1119895 minus 119901119899119909119895Δ1199092

= 2Δ119909 (119901119899119909+1119895 minus119872sum119899=1

119888119899 (120593119899)119899119909119895) (120597119901120597119909)0119895 =

1199011119895 minus 1199010119895Δ1199092= 2Δ119909 (

119872sum119899=1

119888119899 (120593119899)1119895 minus 1199010119895) (120597119901120597119909)119894119899119910+1 =

119901119894119899119910+1 minus 119901119894119899119910Δ1199102= 2Δ119910 (119901119894119899119910+1 minus

119872sum119899=1

119888119899 (120593119899)119894119899119910) (120597119901120597119909)1198940 =

1199011198941 minus 1199011198940Δ1199102= 2Δ119910 (

119872sum119899=1

119888119899 (120593119899)1198941 minus 1199011198940)

(25)

where the boundary pressures 119901119899119909+1119895 1199010119895 119901119894119899119910+1 and 1199011198940are constant so that they remain in the expressions andtheir neighbor pressures (119901119899119909119895 1199011119895 119901119894119899119910 1199011198941) are changingwith time so that they are decomposed into 119888119899 and 120593119899 using

(11) Thus the projection of the boundary condition can betransformed to

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

]

+ 2Δ119909Δ119910119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

+ (119896119910119910120583 1205931198981199012)1198940

]

minus 119872sum119899=1

1198881198992Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910

+ (119896119910119910120583 120593119898119901)1198940

(120593119899)1198941]

(26)

(2) If all the boundaries are Neumann condition (knownboundary velocity) boundary treatment can be expressed asfollows

1199060119895 = minus(119896119909119909120583 120597119901120597119909)0119895 119906119899119909119895 = minus(119896119909119909120583 120597119901120597119909)119899119909+1119895

V1198940 = minus(119896119910119910120583 120597119901120597119910)1198940

V119894119899119910 = minus(119896119910119910120583 120597119901120597119910)119894119899119910+1

(27)

Geofluids 5

The projection of the boundary condition can be transformedto

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119872sum119899=1

119888119899

sdot 119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895] Δ119910

+ 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 minus V119894119899119910 (120593119898120593119899)119894119899119910+1] Δ119909

(28)

(3) For general boundary conditions the projection of theboundary condition can be summarized via both (26) and(28)

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119867119861119898 + 119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

(29)

where

119867119861119898 = 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

Diri119883119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

Diri119884119894119899119910+1+ (119896119910119910120583 1205931198981199012)

1198940

Diri1198841198940]

119867119881119898119899 = Δ119910119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 (1 minus Diri1198830119895)minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895 (1 minus Diri119883119899119909+1119895)]+ Δ119909 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 (1 minus Diri1198841198940)minus V119894119899119910 (120593119898120593119899)119894119899119910+1 (1 minus Diri119884119894119899119910+1)]

119867119875119898119899 = 2Δ119910Δ119909sdot 119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895Diri119883119899119909+1119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910Diri119884119894119899119910+1+ (119896119910119910120583 120593119898119901)

1198940

(120593119899)1198941Diri1198841198940] (30)

where Diri119883 and Diri119884 are 1 for Dirichlet boundary condi-tion and 0 forNeumannboundary conditionThus boundarypressures only appear in the expressions of 119867119861119898 and 119867119875119898119899while boundary velocities only appear in the expression of119867119881119898119899 119867119861119898 119867119881119898119899 and 119867119875119898119899 can also be calculated onlyonce saving computational time Let

119867119880119898119899 = int1198971199100int1198971199090119888119892119901120593119898120593119899119889119909 119889119910

= 119899119910sum119895=1

119899119909sum119894=1

(119888119892119901120593119898120593119899)119894119895 Δ119909Δ119910(31)

119867119863119898119899119897= int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

= 119899119910sum119895=1

119899119909sum119894=1

[120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )]119894119895

Δ119909Δ119910(32)

119867119878119898 = int1198971199100int1198971199090120593119898119885119902119889119909119889119910 =

119899119910sum119895=1

119899119909sum119894=1

(120593119898119885119902)119894119895 Δ119909Δ119910 (33)

Then the POD model for gas reservoir simulation can beobtained in the following expression

120601119872sum119899=1

119889119888119899119889119905 119867119880119898119899 = 119867119861119898 +119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897119867119863119898119899119897 + 119877119879119882 119867119878119898(34)

Equation (34) is the time evolution equation of 119888119899 Foreach time step (34) can be solved via an efficient linearsolver (DLSARG) provided by FORTRAN coding languageNumerical results are discussed in the next section to evaluatethe precision and the acceleration performances of the PODreduced-order model

6 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 1 Computational domain and permeability

3 Results and Discussions

The computational domain (100m times 100m) is shown inFigure 1 Permeability takes the value 119896119904 in the blue area (50mtimes 50m in the center of the domain) and 119896119897 in the red areaNatural gas with the main component of methane (CH4) isthe most important gas in subsurface reservoirs Thus it isused in the numerical cases in this paper Initial pressure 1199010 isgiven in the whole domain with zero injection or production119901lb and 119901rb are imposed on the left and the right boundariesNo flow boundary condition is applied to the top and bottomboundaries All parameters are listed in Table 1 where 1md =9869233 times 10minus16m2

In this condition governing equations (see (4) and(7)sim(10)) are solved to collect samples at different momentsusing the highly accurate finite difference method (FDM)to ensure the grid-independent results for the grid numbermore than 60 times 60 [29] Thus the grid number in this paper(100 times 100) is enough The temporal scheme is an explicitadvancement for each time step (Δ119905 = 1296 s) with 2 times 106time steps After the total time scale of the simulation of 30days 2000 samples are collected To evaluate the precisionand computational speed of the POD model the relativedeviation 120576 and the acceleration ratio 119903 are defined in thefollowing

120576 = 1003817100381710038171003817119901POD minus 119901FDM10038171003817100381710038171003817100381710038171003817119901FDM1003817100381710038171003817 times 100 (35)

119903 = 119905FDM119905POD (36)

where 119905 is the CPU time for the computation and thesubscripts ldquoPODrdquo and ldquoFDMrdquo represent the results from thePODmodel and from the direct calculation of the governingequations via the FDM If 119903 gt 1 the POD model can

Table 1 Computational parameters

Parameter Value Unit120601 02 1199010 101325 Pa119901lb 1013250 Pa119901rb 101325 Pa119902 0 kg(m3sdots)119896119904 1 md119896119897 100 md119877 83147295 J(molsdotK)119879 298 K119882 16 times 10minus3 Kgmol120583 11067 times 10minus6 Pasdots119886 02283 Pasdotm6sdotmolminus2119887 4278 times 10minus5 m3mol119872 2000 119899119909 100 119899119910 100 119897119909 100 m119897119910 100 mΔ119909 1 mΔ119910 1 mΔ119905 1296 sSimulation time scope 30 Days

accelerate the computationThe larger 119903 represents the largeracceleration

31 POD Results for Ideal Gas First of all the POD modelis examined in the case of ideal gas (119885 equiv 1) where 119886 and119887 in the Van der Waals equation (see (4)) are all zero Thusthe expression of the gas compressibility (see (7)) decays to119888119892 = 1119901 so that (31) and (33) can be simplified as

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(120593119898120593119899)119894119895 Δ119909Δ119910

119867119878119898 =119899119910sum119895=1

119899119909sum119894=1

(120593119898119902)119894119895 Δ119909Δ119910(37)

As shown in Figure 2 the first POD mode occupies thevast majority (9686) of the total energy This indicatesthat the first mode captures the main characteristics of thewhole transient process However the relative deviation ofthe reconstruction results largely fluctuates with time Themaximum deviation is as high as 210 while the minimumdeviation is only 46 times 10minus2 (Table 2) This phenomenonindicates that a dominant mode occupying the vast majorityof energy can only obtain high-accurate reconstruction atsome moments but loses the fidelity at other moments Topromote the reconstruction precision of the whole transientprocess more POD modes should be included To ensurehigh acceleration of the reduced-order model the inclusionof the POD modes should be as little as possible Thus there

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

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Submit your manuscripts atwwwhindawicom

Geofluids 3

whereΩmeans domain The matrix form of (14) is

C = Δ119909Δ119910119873 S119879S (15)

The matrix C can be decomposed by eigenvalue decom-position to obtain the eigenvalues and eigenvectors

CV = 120582V (16)

whereV is the eigenvectormatrix and 120582 are the eigenvalues ofthe corresponding eigenvectors Along with the descendingorder of 120582 (1205821 gt 1205822 gt 1205823 gt sdot sdot sdot gt 120582119873) the importanceof eigenvectors V1 V2 V3 V119873 in V is also descending Toquantitatively represent the importance of each eigenvectorthe energy 120585119899 of the 119899th eigenvector is introduced

120585119899 = 120582119899sum119873119894=1 120582119894 times 100 (17)

Correspondingly the cumulative energy 120577119899 of the top 119899eigenvectors can be expressed as follows

120577119899 = sum119899119895=1 120582119895sum119873119894=1 120582119894 times 100 (18)

POD modes 120593119899 can be obtained as follows

120593119899 = SVSV (19)

where ldquordquo means the Euclidean distance

(3) Substituting (11) into (8) and projecting the wholeequation onto the Hilbert space spanned by the POD modes120601119898 (119898 = 1 sim 119872) one can finally obtain a linear equationsystem of 119888119899 in the scale of119872 times119872 where119872 is the numberof POD modes in (11) Once the temporal coefficients 119888119899 aresolved the original variable 119901 can be directly reconstructedin (11) instead of the computation of (8) The computationalcost of (11) is in the scale of119872times119872 while the computationalcost of (8) is in the scale of 119871 times 119871 The number of samples isusually quite smaller than the number of grid points (119873 ≪119871) The number of effective POD modes is smaller than thenumber of samples (119872 le 119873) Thus the computational timecan be largely reduced using POD The POD model is atype of reduced-order model The details of this step will beexplained in Section 23

According to the above analyses the core computationof POD is the calculation of temporal coefficients 119888119899 Theequation of 119888119899 for gas reservoir simulation will be establishedin detail in the next section

23 Establishment of the POD Model for Gas ReservoirSimulation Substituted by (11) (8) can be transformed to

120601119888119892119901119872sum119899=1

120593119899 119889119888119899119889119905= 119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]

+ 119885119877119879119882 119902

(20)

Equation (20) is projected onto 120593119898

int1198971199100int1198971199090120593119898120601119888119892119901119872sum

119899=1120593119899 119889119888119899119889119905 119889119909 119889119910 = int1198971199100 int1198971199090 120593119898

119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910 + int1198971199100 int1198971199090 120593119898119885119877119879119882 119902119889119909119889119910 (21)

where 119897119909 and 119897119910 are the lengths of the computational domainin the 119909 and 119910 directions respectively As stated in (11) 119888119899 and 120593119899 are functions of time and space respectivelyThus the left-

hand side of (21) is transformed to

int1198971199100int1198971199090120593119898120601119888119892119901119872sum

119899=1120593119899 119889119888119899119889119905 119889119909 119889119910 = 120601

119872sum119899=1

119889119888119899119889119905 int1198971199100 int1198971199090 119888119892119901120593119898120593119899119889119909 119889119910 (22)

Using integration by part and the property of the PODmodes (see (12)) we can simplify the first term and

the second term of the right-hand side of (21) as fol-lows

int1198971199100int1198971199090120593119898 119872sum119899=1

119872sum119897=1

119888119899119888119897 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910

= 119872sum119899=1

119872sum119897=1

119888119899119888119897 int1198971199100int1198971199090120593119898 [ 120597120597119909 (119896119909119909120583 120593119897 120597120593119899120597119909 ) + 120597120597119910 (

119896119910119910120583 120593119897 120597120593119899120597119910 )]119889119909119889119910 = int119897119909

0[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

4 Geofluids

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909 + int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897 int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

(23)

int1198971199100int1198971199090120593119898119885119877119879119882 119902119889119909119889119910 = 119877119879119882 int119897119910

0int1198971199090120593119898119885119902119889119909119889119910 (24)

Please note that the projection of the boundary conditionin (23) is dependent on boundary pressures and boundarypressure gradients To update them the pressure field shouldbe firstly reconstructed using (11) in every time step once 119888119899 isupdated by the POD model Therefore the projection of theboundary condition leads to additional computations of thefull-order equations (see (9) (10) and (11)) consumingmuchmore time This is very harmful for the high acceleration ofthe PODmodel To ensure the high-acceleration advantage ofthe PODmodel the computations of (9) (10) and (11) shouldbe avoided in the PODmodelThis can be fulfilled by furtherdecomposing the pressures in (23) The derivations will bemade in the following three steps

(1) If all the boundaries are Dirichlet condition (knownboundary pressure) boundary treatment can be expressed asfollows

(120597119901120597119909)119899119909+1119895 =119901119899119909+1119895 minus 119901119899119909119895Δ1199092

= 2Δ119909 (119901119899119909+1119895 minus119872sum119899=1

119888119899 (120593119899)119899119909119895) (120597119901120597119909)0119895 =

1199011119895 minus 1199010119895Δ1199092= 2Δ119909 (

119872sum119899=1

119888119899 (120593119899)1119895 minus 1199010119895) (120597119901120597119909)119894119899119910+1 =

119901119894119899119910+1 minus 119901119894119899119910Δ1199102= 2Δ119910 (119901119894119899119910+1 minus

119872sum119899=1

119888119899 (120593119899)119894119899119910) (120597119901120597119909)1198940 =

1199011198941 minus 1199011198940Δ1199102= 2Δ119910 (

119872sum119899=1

119888119899 (120593119899)1198941 minus 1199011198940)

(25)

where the boundary pressures 119901119899119909+1119895 1199010119895 119901119894119899119910+1 and 1199011198940are constant so that they remain in the expressions andtheir neighbor pressures (119901119899119909119895 1199011119895 119901119894119899119910 1199011198941) are changingwith time so that they are decomposed into 119888119899 and 120593119899 using

(11) Thus the projection of the boundary condition can betransformed to

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

]

+ 2Δ119909Δ119910119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

+ (119896119910119910120583 1205931198981199012)1198940

]

minus 119872sum119899=1

1198881198992Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910

+ (119896119910119910120583 120593119898119901)1198940

(120593119899)1198941]

(26)

(2) If all the boundaries are Neumann condition (knownboundary velocity) boundary treatment can be expressed asfollows

1199060119895 = minus(119896119909119909120583 120597119901120597119909)0119895 119906119899119909119895 = minus(119896119909119909120583 120597119901120597119909)119899119909+1119895

V1198940 = minus(119896119910119910120583 120597119901120597119910)1198940

V119894119899119910 = minus(119896119910119910120583 120597119901120597119910)119894119899119910+1

(27)

Geofluids 5

The projection of the boundary condition can be transformedto

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119872sum119899=1

119888119899

sdot 119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895] Δ119910

+ 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 minus V119894119899119910 (120593119898120593119899)119894119899119910+1] Δ119909

(28)

(3) For general boundary conditions the projection of theboundary condition can be summarized via both (26) and(28)

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119867119861119898 + 119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

(29)

where

119867119861119898 = 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

Diri119883119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

Diri119884119894119899119910+1+ (119896119910119910120583 1205931198981199012)

1198940

Diri1198841198940]

119867119881119898119899 = Δ119910119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 (1 minus Diri1198830119895)minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895 (1 minus Diri119883119899119909+1119895)]+ Δ119909 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 (1 minus Diri1198841198940)minus V119894119899119910 (120593119898120593119899)119894119899119910+1 (1 minus Diri119884119894119899119910+1)]

119867119875119898119899 = 2Δ119910Δ119909sdot 119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895Diri119883119899119909+1119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910Diri119884119894119899119910+1+ (119896119910119910120583 120593119898119901)

1198940

(120593119899)1198941Diri1198841198940] (30)

where Diri119883 and Diri119884 are 1 for Dirichlet boundary condi-tion and 0 forNeumannboundary conditionThus boundarypressures only appear in the expressions of 119867119861119898 and 119867119875119898119899while boundary velocities only appear in the expression of119867119881119898119899 119867119861119898 119867119881119898119899 and 119867119875119898119899 can also be calculated onlyonce saving computational time Let

119867119880119898119899 = int1198971199100int1198971199090119888119892119901120593119898120593119899119889119909 119889119910

= 119899119910sum119895=1

119899119909sum119894=1

(119888119892119901120593119898120593119899)119894119895 Δ119909Δ119910(31)

119867119863119898119899119897= int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

= 119899119910sum119895=1

119899119909sum119894=1

[120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )]119894119895

Δ119909Δ119910(32)

119867119878119898 = int1198971199100int1198971199090120593119898119885119902119889119909119889119910 =

119899119910sum119895=1

119899119909sum119894=1

(120593119898119885119902)119894119895 Δ119909Δ119910 (33)

Then the POD model for gas reservoir simulation can beobtained in the following expression

120601119872sum119899=1

119889119888119899119889119905 119867119880119898119899 = 119867119861119898 +119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897119867119863119898119899119897 + 119877119879119882 119867119878119898(34)

Equation (34) is the time evolution equation of 119888119899 Foreach time step (34) can be solved via an efficient linearsolver (DLSARG) provided by FORTRAN coding languageNumerical results are discussed in the next section to evaluatethe precision and the acceleration performances of the PODreduced-order model

6 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 1 Computational domain and permeability

3 Results and Discussions

The computational domain (100m times 100m) is shown inFigure 1 Permeability takes the value 119896119904 in the blue area (50mtimes 50m in the center of the domain) and 119896119897 in the red areaNatural gas with the main component of methane (CH4) isthe most important gas in subsurface reservoirs Thus it isused in the numerical cases in this paper Initial pressure 1199010 isgiven in the whole domain with zero injection or production119901lb and 119901rb are imposed on the left and the right boundariesNo flow boundary condition is applied to the top and bottomboundaries All parameters are listed in Table 1 where 1md =9869233 times 10minus16m2

In this condition governing equations (see (4) and(7)sim(10)) are solved to collect samples at different momentsusing the highly accurate finite difference method (FDM)to ensure the grid-independent results for the grid numbermore than 60 times 60 [29] Thus the grid number in this paper(100 times 100) is enough The temporal scheme is an explicitadvancement for each time step (Δ119905 = 1296 s) with 2 times 106time steps After the total time scale of the simulation of 30days 2000 samples are collected To evaluate the precisionand computational speed of the POD model the relativedeviation 120576 and the acceleration ratio 119903 are defined in thefollowing

120576 = 1003817100381710038171003817119901POD minus 119901FDM10038171003817100381710038171003817100381710038171003817119901FDM1003817100381710038171003817 times 100 (35)

119903 = 119905FDM119905POD (36)

where 119905 is the CPU time for the computation and thesubscripts ldquoPODrdquo and ldquoFDMrdquo represent the results from thePODmodel and from the direct calculation of the governingequations via the FDM If 119903 gt 1 the POD model can

Table 1 Computational parameters

Parameter Value Unit120601 02 1199010 101325 Pa119901lb 1013250 Pa119901rb 101325 Pa119902 0 kg(m3sdots)119896119904 1 md119896119897 100 md119877 83147295 J(molsdotK)119879 298 K119882 16 times 10minus3 Kgmol120583 11067 times 10minus6 Pasdots119886 02283 Pasdotm6sdotmolminus2119887 4278 times 10minus5 m3mol119872 2000 119899119909 100 119899119910 100 119897119909 100 m119897119910 100 mΔ119909 1 mΔ119910 1 mΔ119905 1296 sSimulation time scope 30 Days

accelerate the computationThe larger 119903 represents the largeracceleration

31 POD Results for Ideal Gas First of all the POD modelis examined in the case of ideal gas (119885 equiv 1) where 119886 and119887 in the Van der Waals equation (see (4)) are all zero Thusthe expression of the gas compressibility (see (7)) decays to119888119892 = 1119901 so that (31) and (33) can be simplified as

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(120593119898120593119899)119894119895 Δ119909Δ119910

119867119878119898 =119899119910sum119895=1

119899119909sum119894=1

(120593119898119902)119894119895 Δ119909Δ119910(37)

As shown in Figure 2 the first POD mode occupies thevast majority (9686) of the total energy This indicatesthat the first mode captures the main characteristics of thewhole transient process However the relative deviation ofthe reconstruction results largely fluctuates with time Themaximum deviation is as high as 210 while the minimumdeviation is only 46 times 10minus2 (Table 2) This phenomenonindicates that a dominant mode occupying the vast majorityof energy can only obtain high-accurate reconstruction atsome moments but loses the fidelity at other moments Topromote the reconstruction precision of the whole transientprocess more POD modes should be included To ensurehigh acceleration of the reduced-order model the inclusionof the POD modes should be as little as possible Thus there

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

4 Geofluids

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909 + int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897 int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

(23)

int1198971199100int1198971199090120593119898119885119877119879119882 119902119889119909119889119910 = 119877119879119882 int119897119910

0int1198971199090120593119898119885119902119889119909119889119910 (24)

Please note that the projection of the boundary conditionin (23) is dependent on boundary pressures and boundarypressure gradients To update them the pressure field shouldbe firstly reconstructed using (11) in every time step once 119888119899 isupdated by the POD model Therefore the projection of theboundary condition leads to additional computations of thefull-order equations (see (9) (10) and (11)) consumingmuchmore time This is very harmful for the high acceleration ofthe PODmodel To ensure the high-acceleration advantage ofthe PODmodel the computations of (9) (10) and (11) shouldbe avoided in the PODmodelThis can be fulfilled by furtherdecomposing the pressures in (23) The derivations will bemade in the following three steps

(1) If all the boundaries are Dirichlet condition (knownboundary pressure) boundary treatment can be expressed asfollows

(120597119901120597119909)119899119909+1119895 =119901119899119909+1119895 minus 119901119899119909119895Δ1199092

= 2Δ119909 (119901119899119909+1119895 minus119872sum119899=1

119888119899 (120593119899)119899119909119895) (120597119901120597119909)0119895 =

1199011119895 minus 1199010119895Δ1199092= 2Δ119909 (

119872sum119899=1

119888119899 (120593119899)1119895 minus 1199010119895) (120597119901120597119909)119894119899119910+1 =

119901119894119899119910+1 minus 119901119894119899119910Δ1199102= 2Δ119910 (119901119894119899119910+1 minus

119872sum119899=1

119888119899 (120593119899)119894119899119910) (120597119901120597119909)1198940 =

1199011198941 minus 1199011198940Δ1199102= 2Δ119910 (

119872sum119899=1

119888119899 (120593119899)1198941 minus 1199011198940)

(25)

where the boundary pressures 119901119899119909+1119895 1199010119895 119901119894119899119910+1 and 1199011198940are constant so that they remain in the expressions andtheir neighbor pressures (119901119899119909119895 1199011119895 119901119894119899119910 1199011198941) are changingwith time so that they are decomposed into 119888119899 and 120593119899 using

(11) Thus the projection of the boundary condition can betransformed to

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

]

+ 2Δ119909Δ119910119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

+ (119896119910119910120583 1205931198981199012)1198940

]

minus 119872sum119899=1

1198881198992Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910

+ (119896119910119910120583 120593119898119901)1198940

(120593119899)1198941]

(26)

(2) If all the boundaries are Neumann condition (knownboundary velocity) boundary treatment can be expressed asfollows

1199060119895 = minus(119896119909119909120583 120597119901120597119909)0119895 119906119899119909119895 = minus(119896119909119909120583 120597119901120597119909)119899119909+1119895

V1198940 = minus(119896119910119910120583 120597119901120597119910)1198940

V119894119899119910 = minus(119896119910119910120583 120597119901120597119910)119894119899119910+1

(27)

Geofluids 5

The projection of the boundary condition can be transformedto

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119872sum119899=1

119888119899

sdot 119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895] Δ119910

+ 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 minus V119894119899119910 (120593119898120593119899)119894119899119910+1] Δ119909

(28)

(3) For general boundary conditions the projection of theboundary condition can be summarized via both (26) and(28)

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119867119861119898 + 119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

(29)

where

119867119861119898 = 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

Diri119883119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

Diri119884119894119899119910+1+ (119896119910119910120583 1205931198981199012)

1198940

Diri1198841198940]

119867119881119898119899 = Δ119910119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 (1 minus Diri1198830119895)minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895 (1 minus Diri119883119899119909+1119895)]+ Δ119909 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 (1 minus Diri1198841198940)minus V119894119899119910 (120593119898120593119899)119894119899119910+1 (1 minus Diri119884119894119899119910+1)]

119867119875119898119899 = 2Δ119910Δ119909sdot 119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895Diri119883119899119909+1119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910Diri119884119894119899119910+1+ (119896119910119910120583 120593119898119901)

1198940

(120593119899)1198941Diri1198841198940] (30)

where Diri119883 and Diri119884 are 1 for Dirichlet boundary condi-tion and 0 forNeumannboundary conditionThus boundarypressures only appear in the expressions of 119867119861119898 and 119867119875119898119899while boundary velocities only appear in the expression of119867119881119898119899 119867119861119898 119867119881119898119899 and 119867119875119898119899 can also be calculated onlyonce saving computational time Let

119867119880119898119899 = int1198971199100int1198971199090119888119892119901120593119898120593119899119889119909 119889119910

= 119899119910sum119895=1

119899119909sum119894=1

(119888119892119901120593119898120593119899)119894119895 Δ119909Δ119910(31)

119867119863119898119899119897= int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

= 119899119910sum119895=1

119899119909sum119894=1

[120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )]119894119895

Δ119909Δ119910(32)

119867119878119898 = int1198971199100int1198971199090120593119898119885119902119889119909119889119910 =

119899119910sum119895=1

119899119909sum119894=1

(120593119898119885119902)119894119895 Δ119909Δ119910 (33)

Then the POD model for gas reservoir simulation can beobtained in the following expression

120601119872sum119899=1

119889119888119899119889119905 119867119880119898119899 = 119867119861119898 +119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897119867119863119898119899119897 + 119877119879119882 119867119878119898(34)

Equation (34) is the time evolution equation of 119888119899 Foreach time step (34) can be solved via an efficient linearsolver (DLSARG) provided by FORTRAN coding languageNumerical results are discussed in the next section to evaluatethe precision and the acceleration performances of the PODreduced-order model

6 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 1 Computational domain and permeability

3 Results and Discussions

The computational domain (100m times 100m) is shown inFigure 1 Permeability takes the value 119896119904 in the blue area (50mtimes 50m in the center of the domain) and 119896119897 in the red areaNatural gas with the main component of methane (CH4) isthe most important gas in subsurface reservoirs Thus it isused in the numerical cases in this paper Initial pressure 1199010 isgiven in the whole domain with zero injection or production119901lb and 119901rb are imposed on the left and the right boundariesNo flow boundary condition is applied to the top and bottomboundaries All parameters are listed in Table 1 where 1md =9869233 times 10minus16m2

In this condition governing equations (see (4) and(7)sim(10)) are solved to collect samples at different momentsusing the highly accurate finite difference method (FDM)to ensure the grid-independent results for the grid numbermore than 60 times 60 [29] Thus the grid number in this paper(100 times 100) is enough The temporal scheme is an explicitadvancement for each time step (Δ119905 = 1296 s) with 2 times 106time steps After the total time scale of the simulation of 30days 2000 samples are collected To evaluate the precisionand computational speed of the POD model the relativedeviation 120576 and the acceleration ratio 119903 are defined in thefollowing

120576 = 1003817100381710038171003817119901POD minus 119901FDM10038171003817100381710038171003817100381710038171003817119901FDM1003817100381710038171003817 times 100 (35)

119903 = 119905FDM119905POD (36)

where 119905 is the CPU time for the computation and thesubscripts ldquoPODrdquo and ldquoFDMrdquo represent the results from thePODmodel and from the direct calculation of the governingequations via the FDM If 119903 gt 1 the POD model can

Table 1 Computational parameters

Parameter Value Unit120601 02 1199010 101325 Pa119901lb 1013250 Pa119901rb 101325 Pa119902 0 kg(m3sdots)119896119904 1 md119896119897 100 md119877 83147295 J(molsdotK)119879 298 K119882 16 times 10minus3 Kgmol120583 11067 times 10minus6 Pasdots119886 02283 Pasdotm6sdotmolminus2119887 4278 times 10minus5 m3mol119872 2000 119899119909 100 119899119910 100 119897119909 100 m119897119910 100 mΔ119909 1 mΔ119910 1 mΔ119905 1296 sSimulation time scope 30 Days

accelerate the computationThe larger 119903 represents the largeracceleration

31 POD Results for Ideal Gas First of all the POD modelis examined in the case of ideal gas (119885 equiv 1) where 119886 and119887 in the Van der Waals equation (see (4)) are all zero Thusthe expression of the gas compressibility (see (7)) decays to119888119892 = 1119901 so that (31) and (33) can be simplified as

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(120593119898120593119899)119894119895 Δ119909Δ119910

119867119878119898 =119899119910sum119895=1

119899119909sum119894=1

(120593119898119902)119894119895 Δ119909Δ119910(37)

As shown in Figure 2 the first POD mode occupies thevast majority (9686) of the total energy This indicatesthat the first mode captures the main characteristics of thewhole transient process However the relative deviation ofthe reconstruction results largely fluctuates with time Themaximum deviation is as high as 210 while the minimumdeviation is only 46 times 10minus2 (Table 2) This phenomenonindicates that a dominant mode occupying the vast majorityof energy can only obtain high-accurate reconstruction atsome moments but loses the fidelity at other moments Topromote the reconstruction precision of the whole transientprocess more POD modes should be included To ensurehigh acceleration of the reduced-order model the inclusionof the POD modes should be as little as possible Thus there

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

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Submit your manuscripts atwwwhindawicom

Geofluids 5

The projection of the boundary condition can be transformedto

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119872sum119899=1

119888119899

sdot 119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895] Δ119910

+ 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 minus V119894119899119910 (120593119898120593119899)119894119899119910+1] Δ119909

(28)

(3) For general boundary conditions the projection of theboundary condition can be summarized via both (26) and(28)

int1198971199090[(120593119898 119896119910119910120583 119901120597119901120597119910)

119897119910

minus (120593119898 119896119910119910120583 119901120597119901120597119910)0

]119889119909

+ int1198971199100[(120593119898 119896119909119909120583 119901120597119901120597119909)119897119909 minus (120593119898

119896119909119909120583 119901120597119901120597119909)0]119889119910

= 119867119861119898 + 119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

(29)

where

119867119861119898 = 2Δ119910Δ119909119899119910sum119895=1

[(119896119909119909120583 1205931198981199012)119899119909+1119895

Diri119883119899119909+1119895

+ (119896119909119909120583 1205931198981199012)0119895

Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 1205931198981199012)119894119899119910+1

Diri119884119894119899119910+1+ (119896119910119910120583 1205931198981199012)

1198940

Diri1198841198940]

119867119881119898119899 = Δ119910119899119910sum119895=1

[1199060119895 (120593119898120593119899)0119895 (1 minus Diri1198830119895)minus 119906119899119909119895 (120593119898120593119899)119899119909+1119895 (1 minus Diri119883119899119909+1119895)]+ Δ119909 119899119909sum119894=1

[V1198940 (120593119898120593119899)1198940 (1 minus Diri1198841198940)minus V119894119899119910 (120593119898120593119899)119894119899119910+1 (1 minus Diri119884119894119899119910+1)]

119867119875119898119899 = 2Δ119910Δ119909sdot 119899119910sum119895=1

[(119896119909119909120583 120593119898119901)119899119909+1119895

(120593119899)119899119909119895Diri119883119899119909+1119895

+ (119896119909119909120583 120593119898119901)0119895

(120593119899)1119895Diri1198830119895] + 2Δ119909Δ119910sdot 119899119909sum119894=1

[(119896119910119910120583 120593119898119901)119894119899119910+1

(120593119899)119894119899119910Diri119884119894119899119910+1+ (119896119910119910120583 120593119898119901)

1198940

(120593119899)1198941Diri1198841198940] (30)

where Diri119883 and Diri119884 are 1 for Dirichlet boundary condi-tion and 0 forNeumannboundary conditionThus boundarypressures only appear in the expressions of 119867119861119898 and 119867119875119898119899while boundary velocities only appear in the expression of119867119881119898119899 119867119861119898 119867119881119898119899 and 119867119875119898119899 can also be calculated onlyonce saving computational time Let

119867119880119898119899 = int1198971199100int1198971199090119888119892119901120593119898120593119899119889119909 119889119910

= 119899119910sum119895=1

119899119909sum119894=1

(119888119892119901120593119898120593119899)119894119895 Δ119909Δ119910(31)

119867119863119898119899119897= int1198971199100int1198971199090120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )119889119909119889119910

= 119899119910sum119895=1

119899119909sum119894=1

[120593119897 (119896119909119909120583 120597120593119899120597119909 120597120593119898120597119909 + 119896119910119910120583 120597120593119899120597119910 120597120593119898120597119910 )]119894119895

Δ119909Δ119910(32)

119867119878119898 = int1198971199100int1198971199090120593119898119885119902119889119909119889119910 =

119899119910sum119895=1

119899119909sum119894=1

(120593119898119885119902)119894119895 Δ119909Δ119910 (33)

Then the POD model for gas reservoir simulation can beobtained in the following expression

120601119872sum119899=1

119889119888119899119889119905 119867119880119898119899 = 119867119861119898 +119872sum119899=1

119888119899 (119867119881119898119899 minus 119867119875119898119899)

minus 119872sum119899=1

119872sum119897=1

119888119899119888119897119867119863119898119899119897 + 119877119879119882 119867119878119898(34)

Equation (34) is the time evolution equation of 119888119899 Foreach time step (34) can be solved via an efficient linearsolver (DLSARG) provided by FORTRAN coding languageNumerical results are discussed in the next section to evaluatethe precision and the acceleration performances of the PODreduced-order model

6 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 1 Computational domain and permeability

3 Results and Discussions

The computational domain (100m times 100m) is shown inFigure 1 Permeability takes the value 119896119904 in the blue area (50mtimes 50m in the center of the domain) and 119896119897 in the red areaNatural gas with the main component of methane (CH4) isthe most important gas in subsurface reservoirs Thus it isused in the numerical cases in this paper Initial pressure 1199010 isgiven in the whole domain with zero injection or production119901lb and 119901rb are imposed on the left and the right boundariesNo flow boundary condition is applied to the top and bottomboundaries All parameters are listed in Table 1 where 1md =9869233 times 10minus16m2

In this condition governing equations (see (4) and(7)sim(10)) are solved to collect samples at different momentsusing the highly accurate finite difference method (FDM)to ensure the grid-independent results for the grid numbermore than 60 times 60 [29] Thus the grid number in this paper(100 times 100) is enough The temporal scheme is an explicitadvancement for each time step (Δ119905 = 1296 s) with 2 times 106time steps After the total time scale of the simulation of 30days 2000 samples are collected To evaluate the precisionand computational speed of the POD model the relativedeviation 120576 and the acceleration ratio 119903 are defined in thefollowing

120576 = 1003817100381710038171003817119901POD minus 119901FDM10038171003817100381710038171003817100381710038171003817119901FDM1003817100381710038171003817 times 100 (35)

119903 = 119905FDM119905POD (36)

where 119905 is the CPU time for the computation and thesubscripts ldquoPODrdquo and ldquoFDMrdquo represent the results from thePODmodel and from the direct calculation of the governingequations via the FDM If 119903 gt 1 the POD model can

Table 1 Computational parameters

Parameter Value Unit120601 02 1199010 101325 Pa119901lb 1013250 Pa119901rb 101325 Pa119902 0 kg(m3sdots)119896119904 1 md119896119897 100 md119877 83147295 J(molsdotK)119879 298 K119882 16 times 10minus3 Kgmol120583 11067 times 10minus6 Pasdots119886 02283 Pasdotm6sdotmolminus2119887 4278 times 10minus5 m3mol119872 2000 119899119909 100 119899119910 100 119897119909 100 m119897119910 100 mΔ119909 1 mΔ119910 1 mΔ119905 1296 sSimulation time scope 30 Days

accelerate the computationThe larger 119903 represents the largeracceleration

31 POD Results for Ideal Gas First of all the POD modelis examined in the case of ideal gas (119885 equiv 1) where 119886 and119887 in the Van der Waals equation (see (4)) are all zero Thusthe expression of the gas compressibility (see (7)) decays to119888119892 = 1119901 so that (31) and (33) can be simplified as

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(120593119898120593119899)119894119895 Δ119909Δ119910

119867119878119898 =119899119910sum119895=1

119899119909sum119894=1

(120593119898119902)119894119895 Δ119909Δ119910(37)

As shown in Figure 2 the first POD mode occupies thevast majority (9686) of the total energy This indicatesthat the first mode captures the main characteristics of thewhole transient process However the relative deviation ofthe reconstruction results largely fluctuates with time Themaximum deviation is as high as 210 while the minimumdeviation is only 46 times 10minus2 (Table 2) This phenomenonindicates that a dominant mode occupying the vast majorityof energy can only obtain high-accurate reconstruction atsome moments but loses the fidelity at other moments Topromote the reconstruction precision of the whole transientprocess more POD modes should be included To ensurehigh acceleration of the reduced-order model the inclusionof the POD modes should be as little as possible Thus there

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

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Submit your manuscripts atwwwhindawicom

6 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 1 Computational domain and permeability

3 Results and Discussions

The computational domain (100m times 100m) is shown inFigure 1 Permeability takes the value 119896119904 in the blue area (50mtimes 50m in the center of the domain) and 119896119897 in the red areaNatural gas with the main component of methane (CH4) isthe most important gas in subsurface reservoirs Thus it isused in the numerical cases in this paper Initial pressure 1199010 isgiven in the whole domain with zero injection or production119901lb and 119901rb are imposed on the left and the right boundariesNo flow boundary condition is applied to the top and bottomboundaries All parameters are listed in Table 1 where 1md =9869233 times 10minus16m2

In this condition governing equations (see (4) and(7)sim(10)) are solved to collect samples at different momentsusing the highly accurate finite difference method (FDM)to ensure the grid-independent results for the grid numbermore than 60 times 60 [29] Thus the grid number in this paper(100 times 100) is enough The temporal scheme is an explicitadvancement for each time step (Δ119905 = 1296 s) with 2 times 106time steps After the total time scale of the simulation of 30days 2000 samples are collected To evaluate the precisionand computational speed of the POD model the relativedeviation 120576 and the acceleration ratio 119903 are defined in thefollowing

120576 = 1003817100381710038171003817119901POD minus 119901FDM10038171003817100381710038171003817100381710038171003817119901FDM1003817100381710038171003817 times 100 (35)

119903 = 119905FDM119905POD (36)

where 119905 is the CPU time for the computation and thesubscripts ldquoPODrdquo and ldquoFDMrdquo represent the results from thePODmodel and from the direct calculation of the governingequations via the FDM If 119903 gt 1 the POD model can

Table 1 Computational parameters

Parameter Value Unit120601 02 1199010 101325 Pa119901lb 1013250 Pa119901rb 101325 Pa119902 0 kg(m3sdots)119896119904 1 md119896119897 100 md119877 83147295 J(molsdotK)119879 298 K119882 16 times 10minus3 Kgmol120583 11067 times 10minus6 Pasdots119886 02283 Pasdotm6sdotmolminus2119887 4278 times 10minus5 m3mol119872 2000 119899119909 100 119899119910 100 119897119909 100 m119897119910 100 mΔ119909 1 mΔ119910 1 mΔ119905 1296 sSimulation time scope 30 Days

accelerate the computationThe larger 119903 represents the largeracceleration

31 POD Results for Ideal Gas First of all the POD modelis examined in the case of ideal gas (119885 equiv 1) where 119886 and119887 in the Van der Waals equation (see (4)) are all zero Thusthe expression of the gas compressibility (see (7)) decays to119888119892 = 1119901 so that (31) and (33) can be simplified as

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(120593119898120593119899)119894119895 Δ119909Δ119910

119867119878119898 =119899119910sum119895=1

119899119909sum119894=1

(120593119898119902)119894119895 Δ119909Δ119910(37)

As shown in Figure 2 the first POD mode occupies thevast majority (9686) of the total energy This indicatesthat the first mode captures the main characteristics of thewhole transient process However the relative deviation ofthe reconstruction results largely fluctuates with time Themaximum deviation is as high as 210 while the minimumdeviation is only 46 times 10minus2 (Table 2) This phenomenonindicates that a dominant mode occupying the vast majorityof energy can only obtain high-accurate reconstruction atsome moments but loses the fidelity at other moments Topromote the reconstruction precision of the whole transientprocess more POD modes should be included To ensurehigh acceleration of the reduced-order model the inclusionof the POD modes should be as little as possible Thus there

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

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Submit your manuscripts atwwwhindawicom

Geofluids 7

Table 2 Reconstruction deviations using top 1simtop 7 modes

POD modes Top 1 Top 2 Top 3 Top 4 Top 5 Top 6 Top 7120576max 210 84 46 29 27 12 13120576min 46 times 10minus2 43 times 10minus3 24 times 10minus3 19 times 10minus4 57 times 10minus3 47 times 10minus3 69 times 10minus4

1 10 100 10000

102030405060708090

100

96

97

98

99

100

101

POD mode n

n(

)

n(

)

Energy contribution nCumulative energy contribution n

Figure 2 Energy spectrum of the POD modes

is an optimal truncation number of modes Theoretically thenumber of the POD modes can be usually determined whenthe cumulative energy contribution achieves 100 As shownin Figure 2 the cumulative energy contribution 120577119899 achieves100 at and after the 7th PODmode Although themaximumdeviation decreases rapidly with increasing number of PODmodes the maximum deviation is still as high as 13 whenthe top 7 modes are used Therefore more modes should beconsidered in the reduced-order model

Relative deviations along with time using more PODmodes are shown in Figure 3With the increasing number ofPODmodes the deviation converges rapidly whenmore thantop 10 modes are used The maximum deviation decreasesgreatly from 1306 at the top 7 modes to 639 at thetop 10 modes After the top 10 modes the decrease of themaximum deviation becomes much slower (531 for the top20 modes 525 for the top 30 modes and 521 for the top40modes)Theminimumdeviations are all smallThus it canbe considered that the reconstruction results converge at thenumber of top 20modes However the computational time ofthe PODmodel is increasing with increasing number of PODmodes causing the acceleration ratio to be largely influenced(Table 3) The largest drop of the acceleration ratio occursbetween the top 10 modes and top 20 modes More PODmodes cause the dimension of the POD model to increasemuch faster so that the computation of a larger equation sys-tem consumesmuchmore timeTherefore the optimal num-ber of PODmodes needs a balance between the precision andthe acceleration Between the top 10modes and top 20modesthe precision is promoted a little (120576 from 639 to 531) butthe acceleration ratio of the POD model decreases from 24for the top 10 modes to 66 for the top 20 modes so that the

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

(

)t (day)

Top 7 modes( = 686 times 10minus4~1306)

Top 10 modes( = 163 times 10minus3~639)

Top 20 modes( = 117 times 10minus3~531)

Top 30 modes( = 671 times 10minus4~525)

Top 40 modes( = 874 times 10minus4~521)

Figure 3 Reconstruction deviations using different numbers ofPOD modes

number of the top 10modes is an appropriate optimal numberof PODmodes From the comparison it should be noted thatthe theoretical energy criterion is not enough to determinethe optimal number of PODmodesThe balance of precisionand acceleration should be considered This will includemuchmore PODmodes with very small energy contributionbut their actual contributions to precision are important

It can also be seen from Figure 3 that the maximumdeviation (639) and theminimumdeviation (163times10minus3)occur at about 08 days and about 22 days when the top 10modes are used in the POD model The velocity field andpressure field obtained by the POD model are comparedwith those obtained by FDM to examine the reconstructionprecision of local characteristics FromFigure 4 it is clear thatthe two components of Darcy velocity and pressure are allreconstructed well with tiny local deviation From Figure 5reconstructed velocity component V and reconstructed pres-sure agree well with those of FDM but reconstructed velocitycomponent 119906 can only capture part of the features Fortu-nately this maximum deviation only exists in a very narrowrange in Figure 3 The deviation at other moments decreasesrapidlyWe examine the local flow field at the mean deviation(354) in Figure 6 and find that the local distributions of 119906V and 119901 are well reconstructed Thus the POD model usingthe top 10 POD modes can reconstruct the flow field in highprecision for the whole transient process

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

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EcologyInternational Journal of

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Applied ampEnvironmentalSoil Science

Volume 2018

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Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

8 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 4 Flow field comparison for the minimum deviation Black line FDM red dashed line POD

32 POD Results for Real Gas For real gas compressibilityfactor (119885) and compressibility (119888119892) should be computed bythe equation of state (see (4)) and the equation of real gascompressibility (see (7)) POD results for the same case inSection 31 are obtained by solving (34) The reconstructionprecision (120576 = 278 times 10minus3sim639) is as high as that of idealgas (Figure 3) Flow field comparisons are also very similar tothose in Figures 4ndash6 Thus these results are not redundantlyshown here The above PODmodel maintains high precisionfor real gas simulation

However the acceleration ratio is as low as 09 (ie6240 s7256 s inTable 4) whichmeans the PODmodel causes

the simulation to be even slower than FDM This is quitedifferent from the high acceleration ratio for ideal gas (119903 = 24in Table 3) The main difference between the computationsof real gas and ideal gas is that the equation of state (EOS)should be computed for real gas We analyze the differentcontributions of the total computational time for FDM andPOD in Table 4 and find two important phenomena (1)the largest part of the total computational time is mainlyconsumed on EOS for both FDM and POD (92 for FDMand 78 for POD) the EOS cannot be expressed as an explicitfunction of pressure so that it cannot be decomposed by thePOD it can only be calculated locally for both FDMand POD

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 9

Table 3 Computational time using top 7simtop 40 modes

POD modes Top 7 Top 10 Top 20 Top 30 Top 40CPU time of POD 15 s 23 s 85 s 246 s 420 s119903 37 24 66 22 13Note The computational time of FDM is 560 s

Table 4 Computational time analyses in the case of real gas

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 7256 s 5660 s 1596 s 78 22

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 5 Flow field comparison for the maximum deviation Black line FDM red dashed line POD

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

10 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 6 Flow field comparison for the mean deviation Black line FDM red dashed line POD

and thus the main computational time of real gas simulationis not reduced in the PODmodel (2) the computational timefor flow equations in POD (1596 s) is much longer than thatin FDM (493 s) with much higher contribution (22) thanthat of FDM (8) This means the POD model is still slowerthan FDM even not considering the computational time ofEOS This can be explained using the expression of the PODmodel In themodel (see (34)) all terms are constant for timeadvancement except the projection of the unsteady term (see(31)) and the projection of the source term (see (33)) Theyshould be calculated in every time step because they containthe variables changing with time (119888119892 119901 and 119885) This kind ofcalculation largely increases the total computations

The above analyses indicate that the improvement of theacceleration ability of the PODmodel should reduce the timecontribution of EOS and avoid the frequent computations ofthe two projection terms in the POD computation Accordingto these two points we propose a new method EOS isonly calculated once at the initial time Then the initialcompressibility the initial compressibility factor and theinitial pressure are used to approximate the two projectionterms in (31) and (33) which are treated as constants inthe following transient computation of the POD modelThrough this treatment the calculations of EOS and the twoprojection terms in every time step are all avoided for thePOD computation so that the computational time of the POD

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 11

Table 5 Computational time for real gas after improvement

Total CPU time CPU time for EOS CPU time for flowequations

Time contribution ofEOS

Time contribution offlow equation

FDM 6240 s 5747 s 493 s 92 8POD 23 s 002 s 2298 s 009 9991

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Update cg ( = 278 times 10minus3~639)

No update cg ( = 195 times 10minus3~655)

Figure 7 Precision of the POD model for different treatment ofcompressibility

model is expected to be reduced greatly The approximationsof the two projection terms are shown as follows

119867119880119898119899 =119899119910sum119895=1

119899119909sum119894=1

(11988801198921199010120593119898120593119899)119894119895 Δ119909Δ119910119867119878119898 =

119899119910sum119895=1

119899119909sum119894=1

(1205931198981198850119902)119894119895 Δ119909Δ119910(38)

where 1198880119892 1198850 and 1199010 are the compressibility the compress-ibility factor and the pressure in the initial condition (119905 = 0)respectively

Figure 7 shows that the relative deviation of the abovetreatment is almost the same as that updating the compress-ibility every time step indicating that this treatment does notaffect the precision of the PODmodel Table 5 shows that thetotal computational time of the POD model has been largelyreduced because the computational time of EOS and flowequations are all reduced greatly The time decrease of EOSis because the EOS is only calculated onceThe time decreaseof flow equations is because the two projection terms are notcalculated with time advancement of the POD model Theseresults demonstrate that large acceleration ratio (119903 = 6240 s23 s = 271) of POD for real gas simulation can only be achievedwhen the time contribution of EOS is greatly reduced

33 Verification of the Improved PODModel in aMoreRealisticCase The improved POD model in the above section isapplied to a more complex case of real gas flow simulation to

100

80

60

40

20

0100806040200

y(m

)

x (m)

Figure 8 Real gas flow in a more realistic reservoir

0 500000 1000000 1500000 2000000 25000000

1

2

3

4

5

6

7

(

)

t (s)

Figure 9 Precision of the improved POD model

confirm its capability Permeability field is shown in Figure 8where the red area has a permeability 100md and the bluearea has a permeability 1md The large permeability repre-sents themain flow path of the gas in the reservoir such as soilor sandThe small permeability represents themain obstaclessuch as rock Higher boundary pressure on the left border(100 atm) is used because gas pressure is usually high in engi-neering Other parameters are the same as the previous case

As shown in Figure 9 the relative deviations are alsoas low as the previous case The maximum and minimum

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

12 Geofluids

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 10 Flow field comparison at the maximum deviation for real gas Black line FDM red dashed line POD

deviations are 665 and 013 respectively indicating highprecision of the improved PODmodel It is further confirmedin Figures 10 and 11 that the local flow fields are reconstructedwell at themaximum andminimum deviations A little largerdeviation of pressure in Figure 10(c) may be caused by theapproximation of gas compressibility However the mainfeatures of the pressure field are correctwhile the velocity fieldhasmuch smaller deviationThus the precision is acceptable

Table 6 shows that the computational time is reducedfrom 5823 s for FDM to 25 s for PODThe acceleration ratio isas high as 233 Therefore the improved POD model can stillgreatly save computational time for more complex reservoirand higher pressure

Table 6 Computational time comparison for real gas

CPU time of FDM CPU time of POD 1199035823 s 25 s 233

4 Conclusions

Proper orthogonal decomposition is utilized in gas reservoirsimulation to accelerate the simulation speed of gas flow insingle-continuum porous media High-precision reconstruc-tion can be achieved using only 10 POD modes with thedeviations as low as 163 times 10minus3sim639 for ideal gas and

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 13

100

80

60

40

20

0100806040200

y(m

)

x (m)

(a) 119906

100

80

60

40

20

0100806040200

y(m

)

x (m)

(b) V

100

80

60

40

20

0100806040200

y(m

)

x (m)

(c) 119901

Figure 11 Flow field comparison at the minimum deviation for real gas Black line FDM red dashed line POD

195 times 10minus3sim655 for real gas The acceleration ratio is ashigh as 24 for the typical POD model of the ideal gas flowHowever the computational speed of the typical PODmodelof the real gas flow is even slower than FDM Two key pointsfor improving the computational speed of the PODmodel arediscussed and verified

(1) The computation of EOS should be avoided in thesolving process of the POD model because the totalcomputational time is dominated by EOS

(2) POD projection terms containing compressibilityof gas should not be updated in every time step

Otherwise the computational time of flow equationswill also be longer than FDM

According to the two points we proposed a new methodto approximate the projection terms in all time steps usingthe initial compressibility so that EOS only needs to becalculated once at the initial condition After this treatmentit is verified in two different cases that the computationalspeed of the POD model is largely promoted while highprecision is retained (013sim665)The computational timeof the real gas reservoir simulation is reduced from6240 s and5823 s of FDM to 23 s and 25 s of POD for the two cases Theacceleration ratios are 271 and 233 respectively

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

14 Geofluids

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work presented in this paper has been supported bythe National Natural Science Foundation of China (NSFC)(nos 51576210 51325603 and 51476073) and the ScienceFoundation of China University of Petroleum Beijing (nos2462015BJB03 2462015YQ0409 and C201602) This work isalso supported by the Project of Construction of InnovativeTeams and Teacher Career Development for Universities andColleges Under Beijing Municipality (no IDHT20170507)and the Foundation of Key Laboratory of Thermo-Fluid Sci-ence and Engineering (Xirsquoan JiaotongUniversity)Ministry ofEducation Xirsquoan 710049 China (KLTFSE2015KF01)

References

[1] H Darcy Les Fontaines Publiques de la Vill de Dijon DalmontParis France 1856

[2] Z Chen G Huan and Y Ma Computational Methods forMultiphase Flows in Porous Media Society for Industrial andApplied Mathematics Philadelphia Pa USA 2006

[3] J Chen and Z Chen ldquoThree-dimensional superconvergentgradient recovery on tetrahedral meshesrdquo International Journalfor Numerical Methods in Engineering vol 108 no 8 pp 819ndash838 2016

[4] J Moortgat S Sun and A Firoozabadi ldquoCompositional mod-eling of three-phase flow with gravity using higher-order finiteelement methodsrdquo Water Resources Research vol 47 no 5Article IDW05511 pp 1ndash26 2011

[5] S Sun and J Liu ldquoA locally conservative finite element methodbased on piecewise constant enrichment of the continuousgalerkin methodrdquo SIAM Journal on Scientific Computing vol31 no 4 pp 2528ndash2548 2009

[6] S Sun and M F Wheeler ldquoDiscontinuous Galerkin methodsfor simulating bioreactive transport of viruses in porousmediardquoAdvances in Water Resources vol 30 no 6-7 pp 1696ndash17102007

[7] S Sun and M F Wheeler ldquoAnalysis of discontinuous Galerkinmethods for multicomponent reactive transport problemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 52 no 5 pp 637ndash650 2006

[8] S Sun and M F Wheeler ldquoA dynamic adaptive locally con-servative and nonconforming solution strategy for transportphenomena in chemical engineeringrdquo Chemical EngineeringCommunications vol 193 no 12 pp 1527ndash1545 2006

[9] L Luo B Yu J Cai and X Zeng ldquoNumerical simulation oftortuosity for fluid flow in two-dimensional pore fractal modelsof porous mediardquo Fractals vol 22 no 4 Article ID 14500152014

[10] WWei J Cai X Hu et al ldquoA numerical study on fractal dimen-sions of current streamlines in two-dimensional and three-dimensional pore fractal models of porousmediardquo Fractals vol23 no 1 Article ID 1540012 2015

[11] V Chandra A Barnett P Corbett et al ldquoEffective integrationof reservoir rock-typing and simulation using near-wellbore

upscalingrdquoMarine and Petroleum Geology vol 67 pp 307ndash3262015

[12] V Gholami and S DMohaghegh ldquoFuzzy upscaling in reservoirsimulation An improved alternative to conventional tech-niquesrdquo Journal of Natural Gas Science and Engineering vol 3no 6 pp 706ndash715 2011

[13] M G Correia C Maschio J C von Hohendorff Filho and D JSchiozer ldquoThe impact of time-dependent matrix-fracture fluidtransfer in upscaling match proceduresrdquo Journal of PetroleumScience and Engineering vol 146 pp 752ndash763 2016

[14] S Gholinezhad S Jamshidi and A Hajizadeh ldquoQuad-treedecomposition method for areal upscaling of heterogeneousreservoirs application to arbitrary shaped reservoirsrdquo Fuel vol139 pp 659ndash670 2015

[15] A Q Raeini M J Blunt and B Bijeljic ldquoDirect simulationsof two-phase flow on micro-CT images of porous media andupscaling of pore-scale forcesrdquo Advances in Water Resourcesvol 74 pp 116ndash126 2014

[16] L S K Fung M O Sindi and A H Dogru ldquoMultiparadigmparallel acceleration for reservoir simulationrdquo SPE Journal vol19 no 4 pp 716ndash725 2014

[17] G Yu B Yu D Han and L Wang ldquoUnsteady-state thermalcalculation of buried oil pipeline using a proper orthogonaldecomposition reduced-order modelrdquo Applied Thermal Engi-neering vol 51 no 1-2 pp 177ndash189 2013

[18] D Han B Yu Y Wang Y Zhao and G Yu ldquoFast thermalsimulation of a heated crude oil pipelinewith a BFC-Based PODreduced-ordermodelrdquoAppliedThermal Engineering vol 88 pp217ndash229 2015

[19] YWang B Yu XWu and PWang ldquoPOD andwavelet analyseson the flow structures of a polymer drag-reducing flow based onDNS datardquo International Journal of Heat andMass Transfer vol55 no 17-18 pp 4849ndash4861 2012

[20] YWang B Yu Z CaoW Zou andG Yu ldquoA comparative studyof POD interpolation and POD projectionmethods for fast andaccurate prediction of heat transfer problemsrdquo InternationalJournal of Heat and Mass Transfer vol 55 no 17-18 pp 4827ndash4836 2012

[21] Y Wang B Yu and S Sun ldquoFast prediction method for steady-state heat convectionrdquo Chemical Engineering amp Technology vol35 no 4 pp 668ndash678 2012

[22] M Ghommem VM Calo and Y Efendiev ldquoMode decomposi-tion methods for flows in high-contrast porous media a globalapproachrdquo Journal of Computational Physics vol 257 pp 400ndash413 2014

[23] M Ghommem M Presho V M Calo and Y EfendievldquoMode decomposition methods for flows in high-contrastporousmedia global-local approachrdquo Journal of ComputationalPhysics vol 253 pp 226ndash238 2013

[24] Y Efendiev J Galvis and E Gildin ldquoLocal-global multiscalemodel reduction for flows in high-contrast heterogeneousmediardquo Journal of Computational Physics vol 231 no 24 pp8100ndash8113 2012

[25] M J Patel E F May and M L Johns ldquoHigh-fidelity reservoirsimulations of enhanced gas recovery with supercritical CO2rdquoEnergy vol 111 pp 548ndash559 2016

[26] W Shen Y Xu X Li W Huang and J Gu ldquoNumericalsimulation of gas and water flow mechanism in hydraulicallyfractured shale gas reservoirsrdquo Journal of Natural Gas Scienceand Engineering vol 35 pp 726ndash735 2016

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 15

[27] X Peng M Wang Z Du B Liang and F Mo ldquoUpscaling andsimulation of composite gas reservoirs with thin-interbeddedshale and tight sandstonerdquo Journal of Natural Gas Science andEngineering vol 33 pp 854ndash866 2016

[28] D A Wood W Wang and B Yuan ldquoAdvanced numerical sim-ulation technology enabling the analytical and semi-analyticalmodeling of natural gas reservoirsrdquo Journal of Natural GasScience and Engineering vol 26 pp 1442ndash1451 2015

[29] YWang and S Sun ldquoDirect calculation of permeability by high-accurate finite difference and numerical integration methodsrdquoCommunications in Computational Physics vol 20 no 2 pp405ndash440 2016

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom