Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 384246 7 pageshttpdxdoiorg1011552013384246

Research ArticleNumerical Solution of a Moving Boundary Problemof One-Dimensional Flow in Semi-Infinite Long PorousMedia with Threshold Pressure Gradient

Jun Yao1 Wenchao Liu1 and Zhangxin Chen2

1 School of Petroleum Engineering China University of Petroleum (East China) Qingdao 266580 China2Department of Chemical and Petroleum Engineering Schulich School of EngineeringUniversity of CalgaryCalgary AB Canada T2N 1N4

Correspondence should be addressed to Jun Yao rcogfr upc126com

Received 15 July 2013 Revised 28 September 2013 Accepted 18 October 2013

Academic Editor Kue-Hong Chen

Copyright copy 2013 Jun Yao et al This 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

A numerical method is presented for the solution of a moving boundary problem of one-dimensional flow in semi-infinite longporousmediawith threshold pressure gradient (TPG) for the case of a constant flow rate at the inner boundary In order to overcomethe difficulty in the space discretization of the transient flow region with a moving boundary in the process of numerical solutionthe system of partial differential equations for the moving boundary problem is first transformed equivalently into a closed systemof partial differential equations with fixed boundary conditions by a spatial coordinate transformation methodThen a stable fullyimplicit finite difference method is adopted to obtain its numerical solution Finally numerical results of transient distance of themoving boundary transient production pressure of wellbore and formation pressure distribution are compared graphically withthose from a published exact analytical solution under different values of dimensionless TPG as calculated from actual experimentaldata Comparison analysis shows that numerical solutions are in good agreement with the exact analytical solutions and there isa big difference of model solutions between Darcyrsquos flow and the fluid flow in porous media with TPG especially for the case of alarge dimensionless TPG

1 Introduction

With the increase of international oil price unconven-tional reservoirs such as low-permeable reservoirs heavy oilreservoirs and shale gas reservoirs [1] have become newdevelopment targets in the field of petroleum engineeringin modern times The relevant research on the kinematicprinciples of the fluid flow in unconventional reservoirs isa recently hot topic Abundant experimental and theoreticalanalyses [2ndash15] have demonstrated that the fluid flow inlow-permeable reservoirs and the Bingham non-Newtonianfluid flow in heavy oil reservoirs do not obey the conven-tional Darcyrsquos law there exists a threshold pressure gradient(TPG) or the yield stress That is the fluid flow happensonly if the formation pressure gradient exceeds the TPGor the shear stress for the fluid is larger than the yieldstress

Many literatures have demonstrated the significance oftaking into account the TPG for the fluid flow in uncon-ventional reservoirs For example Zhu et al [16] conductedexperimental investigation of gas flow in water-bearing tightgas reservoirs with TPG and analytical investigation of math-ematical model of low-velocity gas flow their calculationresults indicate that the peripheral reserves of the wellboreare difficult to deploy and the reservoir energy is mainlyconsumednear thewellbore due to existence of TPGwhich isreally unlike Darcyrsquos flow Zhu et al [17] presented a methodfor improving history-matching precision of low-permeablereservoir numerical simulation by considering TPG andrelevant applications in Units X10 and X11 of Daqing Oilfieldverify the obviously improved history-matching precisionof reservoir numerical simulation Yin and Pu [18] per-formed study on the surfactant flooding simulation for low-permeable oilfield in the condition of TPG the enhanced

2 Mathematical Problems in Engineering

matching degree between theoreticalmodel and field practicewas verified through a pilot test of surfactant flooding inChao45 Block of Daqing Oilfield

Due to the derivative discontinuity for the modifiedDarcyrsquos law [2] the mathematical model for this physicswhich the oilfield-development technologies such as welltesting and reservoir numerical simulation for unconven-tional reservoirs involve should be built as a moving bound-ary problem Some analytical or numerical investigations[19ndash30] have demonstrated the big difference between themathematical models with considering moving boundaryconditions and those without considering moving boundaryconditions First of all pressure distribution curves calcu-lated from the mathematical model with moving boundaryconditions due to the TPG show compact support [29 30]which is similar to the characteristics of power-law non-Newtonian fluid flow [20 22 25] in porousmedia or the fluidflow in nanoporous media [1] Wu et al [20] incorporatedmoving boundary conditions for theoretical study of theradial flow and displacement of a Bingham fluid in porousmedia with TPG by integral method and also presented aconfirmed reservoir well-test-analysis method Feng et al[24] studied the model of unsteady radial flow in low-permeable gas reservoirs considering theTPGbyusingGreenfunction method with numerical approximation the movingboundary can represent the single-well control radius byusing the data from Sichuan Gas Field computation resultsshow that the mathematical model with moving boundaryconditions could correctly reflect the seepage mechanics andproduction performance for low-permeable gas reservoirsWang et al [27 28] studied the effect of moving boundaryon the transient radial flow in low-permeable reservoirs withTPG and also demonstrated the significance of taking intoaccount the effect of the moving boundary due to the TPGfor engineering applications

The moving boundary problem of fluid flow in porousmedia with TPG is different from the classical Stefan problemin heat conduction theory [29] the velocity of the movingboundary is proportional to the second derivative of theunknown pressure function with respect to the distance atthis moving boundary However due to the strong nonlin-earity of the moving boundary problem it is very difficult toobtain its exact analytical solution whereas recently in thenew published paper [29] exact analytical solutions for thenon-Stefan moving boundary problems of one-dimensionalflow in semi-infinite long porous media with TPG are pre-sented through a similarity transformation method Beforethe occurrence of the exact analytical solutions approximateanalytical solutions [19ndash21 26] and numerical solutions [22ndash25 27 28 30] have been the main research tools for solvingthe non-Stefan moving boundary problems of the fluidflow in porous media with TPG However these solutionsgenerally lack a strict verification with the related exactanalytical solutions Furthermore some problems of fluidflow in porous media with threshold pressure are muchmore complicated and nonlinear such as the considerationof nonlinear terms in the governing equations [31] and thecoupling of stress sensitive effect in low-permeable porousmedia [32] and then it will be extremely hard to obtain

the exact analytical solutions Therefore it becomes verynecessary to develop a verified numerical method by thepublished exact analytical solution [29] in order to solvemore complicated moving boundary problems of fluid flowin porous media with TPG

The objective of this paper is to present a simple andnovel method for numerical solution of the moving bound-ary problem of one-dimensional flow in semi-infinite longporous media with TPG for the case of a constant flow rateat the inner boundary First this moving boundary problemis equivalently transformed into a closed partial differentialequation system with fixed boundary conditions by thespatial coordinate transformation method [33ndash35] Then astable fully implicit finite difference method [36] is adoptedto obtain its numerical solution Finally the accuracy of thenumerical solution is verified through graphically comparingwith the published exact analytical solution The numericalmethod presented here is applicable to the moving boundaryproblems of multidimensional flow in porous media withTPG

2 Mathematical Model

The problem considered [29] involves the one-dimensionalflow in a semi-infinite long porous mediumwith TPG for thecase of a constant flow rate at the inner boundary the porousmedium is homogeneous isotropic and isothermal thesingle-phase horizontal flow does not have any gravity effectand the fluid and porous medium are slightly compressible

The continuous (mass balance) equation for the one-dimensional flow in the porous medium is given as follows[20 21 26 29]

1205972119875119863

1205971199092119863

=120597119875119863

120597119905119863

0 le 119909119863le 120575 (119905

119863) (1)

where 119875119863is the dimensionless formation pressure 119909

119863is the

dimensionless distance 119905119863is the dimensionless time and 120575 is

the dimensionless distance of the moving boundaryThe initial conditions are as follows

119875119863

1003816100381610038161003816119905119863=0= 0 (2)

120575 (0) = 0 (3)

The inner boundary conditionwith a constant productionrate is as follows

120597119875119863

120597119909119863

10038161003816100381610038161003816100381610038161003816119909119863=0

= minus (1 + 120582119863) (4)

where 120582119863is the dimensionless TPG

The moving boundary conditions are as follows [20 2126 29]

120597119875119863

120597119909119863

10038161003816100381610038161003816100381610038161003816119909119863=120575(119905119863)

= minus120582119863 (5)

119875119863

1003816100381610038161003816119909119863=120575(119905119863)= 0 (6)

Mathematical Problems in Engineering 3

Equations (1)ndash(6) form the mathematical model for theone-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary from (1) (5) and (6) the velocity of the movingboundary can be deduced as follows [29]

120597120575

120597119905119863

=1

120582119863

sdot1205972119875119863

1205971199092119863

100381610038161003816100381610038161003816100381610038161003816119909119863=120575(119905119863)

(7)

3 Numerical Solution of the Problem

It is well known that numerical solution needs the spacediscretization for the flow region however due to theexistence of the moving boundary in the model the one-dimensional flow region is not fixed but expands outsidecontinuously with the time increasing according to (7) Inorder to overcome the difficulty in the space discretizationfor the transient flow region with the moving boundary aspatial coordinate transformation method is introduced [33ndash35]Then themathematicalmodel with themoving boundaryconditions can be transformed into a mathematical modelwith fixed boundary conditions The formula of the spatialcoordinate transformation is as follows

119910119863(119909119863 119905119863) =

119909119863

120575 (119905119863) 0 le 119909

119863le 120575 (119905

119863) (8)

By the spatial coordinate transformation the followingtwo end points in the one-dimensional spatial coordinate of119909119863can be respectively transformed as

119910119863(0 119905119863) =

0

120575 (119905119863)= 0

119910119863(120575 (119905119863) 119905119863) =

120575 (119905119863)

120575 (119905119863)= 1

(9)

By (8) the dimensionless pressure 119875119863(119909119863 119905119863) at the flow

interval of the moving boundary model [0 120575(119905119863)] in the

one-dimensional spatial coordinate of 119909119863can be equivalently

transformed to the function 119863(119910119863 119905119863) at the fixed interval

[0 1] in the one-dimensional spatial coordinate of 119910119863 and

then the differential variables in the mathematical model canbe transformed respectively as follows

120597119875119863

120597119909119863

=120597119863

120597119910119863

sdot1

120575 (10)

1205972119875119863

1205971199092119863

=1205972119863

1205971199102119863

sdot1

1205752 (11)

120597119875119863

120597119905119863

=120597119863

120597119905119863

+120597119863

120597119910119863

sdot (minus119909119863

1205752) sdot

120597120575

120597119905119863

=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575

(12)

Substituting (10)ndash(12) into (1) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575 0 le 119910

119863le 1 (13)

Substituting (10)ndash(12) into (4)ndash(7) respectively yields

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=0

= minus (1 + 120582119863) (14)

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= minus120582119863 (15)

119863

10038161003816100381610038161003816119910119863=1= 0 (16)

120597120575

120597119905119863

=1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot1

1205752 (17)

Substituting (17) into (13) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot119910119863

1205753 (18)

that is

1205972119863

1205971199102119863

sdot 120575 =120597119863

120597119905119863

sdot 1205753minus120597119863

120597119910119863

sdot119910119863

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(19)

The equivalent form of (14) is as follows

120575 = minus120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

sdot1

1 + 120582119863

(20)

Substituting (20) into (19) to cancel the variable 120575 yields

1205972119863

1205971199102119863

sdot120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

minus1

(1 + 120582119863)2(120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

)

3

sdot120597119863

120597119905119863

minus1 + 120582119863

120582119863

sdot 119910119863sdot120597119863

120597119910119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= 0

(21)

From (8) (2) can be transformed as

119863(119910119863 119905119863)10038161003816100381610038161003816119905119863=0

= 0 (22)

From (14) and (15) the following equation can bededuced

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

=1 + 120582119863

120582119863

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(23)

Equations (21)ndash(23) and (16) form a closed system ofpartial differential equations with the fixed boundary condi-tions with respect to

119863(119910119863 119905119863) which is equivalent to the

mathematical model for the moving boundary problem ofthe one-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary From (21) it can be seen that the partial differentialequations have strong nonlinearity which indirectly reflects

4 Mathematical Problems in Engineering

the strong nonlinearity of the original moving boundarymodel Owing to the strong nonlinearity of the transformedmodel a stable fully implicit finite difference method [36] isadopted to numerically solve the relevant nonlinear partialdifferential equations

The one-dimensional unit interval is discretized as 119873

spatial grid subintervals with the same length and then thelength of every subinterval Δ119910

119863is equal to 1119873 the distance

at the 119894th spatial grid 119910119863119894

is 119894 sdot Δ119910119863 the time step is assumed

to be Δ119905119863 replace the first derivative by first-order forward

difference and the second derivative by second-order centraldifference and finally the fully implicit difference equations[36] corresponding to (21) can be expressed as follows

119895+1

119863119894+1minus 2119895+1

119863119894+ 119895+1

119863119894minus1

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119894minus 119895

119863119894

Δ119905119863

minus1 + 120582119863

120582119863

sdot 119894 sdot Δ119910119863sdot

119895+1

119863119894+1minus 119895+1

119863119894

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(119894 = 1 2 119873 minus 2)

(24)

where 119895 denotes the index of time stepThe difference equation of (16) is as follows

119895+1

119863119873= 0 (25)

Then by (25) the difference equation corresponding tothe (119873minus1)th spatial grid can be expressed as follows

minus2119895+1

119863119873minus1+ 119895+1

119863119873minus2

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119873minus1minus 119895

119863119873minus1

Δ119905119863

+1 + 120582119863

120582119863

sdot (119873 minus 1) sdot Δ119910119863sdot

119895+1

119863119873minus1

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(26)

The difference equation of (23) is as follows

119895+1

1198631minus 119895+1

1198630

Δ119910119863

= minus1 + 120582119863

120582119863

119895+1

119863119873minus1

Δ119910119863

(27)

From (22) the initial values of the unknown dimension-less formation pressure are as follows

0

119863119894= 0 119894 = 0 1 2 119873 minus 1 (28)

From (24) (26) and (27) it can be seen that there are119873difference equations at the (119895 + 1)th time step which contain

119873 unknown variables 119895+1119863119894

(119894 = 0 1 2 119873 minus 1) due to thestrong nonlinearity of these difference equations theNewton-Raphson iteration method [36] is adopted to obtain theirnumerical solutions when

119895+1

119863119894(119894 = 0 1 2 119873 minus 1) are

numerically solved for let 119895 be equal to 119895 + 1 and in the samemanner the numerical solutions of 119873 difference equationswith respect to119873 unknown variables 119895+2

119863119894(119894 = 0 1 2 119873minus

1) at the (119895 + 2)th time step can also be numerically solvedfor the rest can be deduced by analogy then the numericalsolutions of the nonlinear partial differential equations withrespect to

119863(119910119863 119905119863) can be obtained

The difference equation of (8) is

119909119895+1

119863119894

= 119910119863119894

sdot 120575119895+1

(29)

The difference equation of (20) is

120575119895+1

= minus

119895+1

1198631minus 119895+1

1198630

Δ119910119863

sdot1

1 + 120582119863

(30)

Substituting (30) into (29) yields

119909119895+1

119863119894

= minus119894 sdot

119895+1

1198631minus 119895+1

1198630

1 + 120582119863

(31)

From (31) in the process of numerical solutions at everytime step the one-dimensional spatial coordinate of 119910

119863can

be transformed as the one of 119909119863 and then the numerical

solutions of 119863(119910119863 119905119863) can be transformed as the numerical

solutions of 119875119863(119909119863 119905119863) From (31) it can also be seen that

it is just the spatial coordinate transformation that is (8)that lets the time-dependent space discretization in the spatialcoordinate of119909

119863be transformed as a time-independent space

discretization in the spatial coordinate of 119910119863 which makes

the numerical solutions by the finite difference method moreapplicable and simple

4 Verification of Numerical Solutions

41 Exact Analytical Solutions The exact analytical solutionof the mathematical model for the one-dimensional flow insemi-infinite long porous media with TPG for the case of aconstant flow rate at the inner boundary is as follows [29]

119875119863(119909119863 119905119863)

= 2 (1 + 120582119863)

times [120579(119905119863

12119890minus1199092

1198634119905119863+119909

119863212058712 erf (119909

1198632radic119905

119863)

119890minus1205792

+12058712120579 erf (120579))minus

119909119863

2]

119909119863isin [0 120575]

(32)

where 120579 can be determined by the following equation withrespect to the TPG [29]

119890minus1205792

119890minus1205792

+ 12058712120579 erf (120579)=

120582119863

1 + 120582119863

(33)

Mathematical Problems in Engineering 5

Table 1 Calculation of dimensionless TPG 120582119863through experimental data [2] of eight samples

Sample 120582

(psicm)119896120583

(mdcp)Minimum flow

rate 1199021(cm3min)

Maximum flowrate 119902

2(cm3min)

Cross-sectionalarea 119860 (cm2)

Average flow velocity]119908= (1199021+ 1199022)1198602

(cmmin)120582119863= 120582119896120583]

119908

Brown sandstone 1 000846 6520 883 30 202 0961139 0242Brown sandstone 2 0014 4810 1319 3232 202 1126485 0252Brown sandstone 3 00203 4730 171 175 202 0475495 0852Brown sandstone 4 0017 3130 579 2046 202 0649752 0345Sandpack 1 00046 27900 381 343 114 1671491 0324Sandpack 2 000789 23500 247 2034 114 1000439 0782Sandpack 3 000548 23800 457 2992 114 1512719 0364Shaly sandstone 0813 566 2 6 114 0350877 0553

It has been proven [29] that for a given value of 120582119863 there

is one and only one positive root of (33)The distance of the moving boundary can be expressed as

follows [29]

120575 = 2 sdot 120579 sdot radic119905119863 (34)

When 120582119863= 0 the exact analytical solution of the mathe-

matical model for the one-dimensional Darcyrsquos flow in semi-infinite long porous media for the case of a constant flow rateat the inner boundary is as follows [29]

119875119863(119909119863 119905119863) = minus 119909

119863+ 2radic

119905119863

120587exp(

minus1199092

119863

4119905119863

)

+ 119909119863erf ( 119909

119863

2radic119905119863

) 119909119863isin [0infin]

(35)

42 Assignment of the Values of Dimensionless TPG 120582119863 The

formula for the dimensionless TPG 120582119863is as follows [29]

120582119863=

119896

120583

120582

]119908

(36)

where 119896 is the permeability of porous media 120582 is the TPG 120583is the fluid viscosity and ]

119908is the average seepage velocity

Here selected Prada and Civanrsquo s actual experimentaldata of eight samples [2] for measuring the one-dimensionalflow in three types of porous media (brown sandstone sand-pack and shaly sandstone) with TPG is used to calculate thevalues of 120582

119863 the experimental data and specific calculation

process are shown in Table 1From Table 1 it can be concluded that the range of the

values of dimensionless TPG 120582119863is between 0242 and 0852

for the eight samples of three types of porous mediaWithoutloss of generality the values of dimensionless TPG 120582

119863are

set 0242 0553 and 0852 for the following numerical testsrespectively

43 Numerical Tests By the numerical method presented inthe paper the mathematical model for the one-dimensionalflow in semi-infinite long porousmedia with TPG for the caseof a constant flow rate at the inner boundary is numerically

00 20 40 60 80 10tD

200

160

120

80

40

0

120575(tD

)

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0852 0553 0242

103

Figure 1 Comparison of transient distance of moving boundaryunder different values of dimensionless TPG

solved where 119873 = 160 Δ119905119863= 10 and 120582

119863= 0242 0553 and

0852 respectively By (33) the values of 120579 correspondingto the three values of 120582

119863can be computed by Newton-

Raphson iteration method [36] as 1005 07732 and 06599for the exact analytical solutions In order to make clear thedifference of model solutions between Darcyrsquos flow when120582119863= 0 and the fluid flow in porous media with the TPG

exact analytical solutions for the Darcyrsquos flow that is (35) arealso added for the purpose of comparison analysis

Under different values of 120582119863 the numerical results are

graphically compared with those from the exact analyticalsolution that is (32)ndash(34) Figures 1ndash3 show the compari-son curves between the numerical solutions and the exactanalytical solutions with respect to the transient distance ofthe moving boundary the transient production pressure ofwellbore and the formation pressure distribution when 119905

119863=

5000 respectivelyFrom Figures 1ndash3 it can be seen that the numerical

solutions are in good agreement with the exact analytical

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

matching degree between theoreticalmodel and field practicewas verified through a pilot test of surfactant flooding inChao45 Block of Daqing Oilfield

Due to the derivative discontinuity for the modifiedDarcyrsquos law [2] the mathematical model for this physicswhich the oilfield-development technologies such as welltesting and reservoir numerical simulation for unconven-tional reservoirs involve should be built as a moving bound-ary problem Some analytical or numerical investigations[19ndash30] have demonstrated the big difference between themathematical models with considering moving boundaryconditions and those without considering moving boundaryconditions First of all pressure distribution curves calcu-lated from the mathematical model with moving boundaryconditions due to the TPG show compact support [29 30]which is similar to the characteristics of power-law non-Newtonian fluid flow [20 22 25] in porousmedia or the fluidflow in nanoporous media [1] Wu et al [20] incorporatedmoving boundary conditions for theoretical study of theradial flow and displacement of a Bingham fluid in porousmedia with TPG by integral method and also presented aconfirmed reservoir well-test-analysis method Feng et al[24] studied the model of unsteady radial flow in low-permeable gas reservoirs considering theTPGbyusingGreenfunction method with numerical approximation the movingboundary can represent the single-well control radius byusing the data from Sichuan Gas Field computation resultsshow that the mathematical model with moving boundaryconditions could correctly reflect the seepage mechanics andproduction performance for low-permeable gas reservoirsWang et al [27 28] studied the effect of moving boundaryon the transient radial flow in low-permeable reservoirs withTPG and also demonstrated the significance of taking intoaccount the effect of the moving boundary due to the TPGfor engineering applications

The moving boundary problem of fluid flow in porousmedia with TPG is different from the classical Stefan problemin heat conduction theory [29] the velocity of the movingboundary is proportional to the second derivative of theunknown pressure function with respect to the distance atthis moving boundary However due to the strong nonlin-earity of the moving boundary problem it is very difficult toobtain its exact analytical solution whereas recently in thenew published paper [29] exact analytical solutions for thenon-Stefan moving boundary problems of one-dimensionalflow in semi-infinite long porous media with TPG are pre-sented through a similarity transformation method Beforethe occurrence of the exact analytical solutions approximateanalytical solutions [19ndash21 26] and numerical solutions [22ndash25 27 28 30] have been the main research tools for solvingthe non-Stefan moving boundary problems of the fluidflow in porous media with TPG However these solutionsgenerally lack a strict verification with the related exactanalytical solutions Furthermore some problems of fluidflow in porous media with threshold pressure are muchmore complicated and nonlinear such as the considerationof nonlinear terms in the governing equations [31] and thecoupling of stress sensitive effect in low-permeable porousmedia [32] and then it will be extremely hard to obtain

the exact analytical solutions Therefore it becomes verynecessary to develop a verified numerical method by thepublished exact analytical solution [29] in order to solvemore complicated moving boundary problems of fluid flowin porous media with TPG

The objective of this paper is to present a simple andnovel method for numerical solution of the moving bound-ary problem of one-dimensional flow in semi-infinite longporous media with TPG for the case of a constant flow rateat the inner boundary First this moving boundary problemis equivalently transformed into a closed partial differentialequation system with fixed boundary conditions by thespatial coordinate transformation method [33ndash35] Then astable fully implicit finite difference method [36] is adoptedto obtain its numerical solution Finally the accuracy of thenumerical solution is verified through graphically comparingwith the published exact analytical solution The numericalmethod presented here is applicable to the moving boundaryproblems of multidimensional flow in porous media withTPG

2 Mathematical Model

The problem considered [29] involves the one-dimensionalflow in a semi-infinite long porous mediumwith TPG for thecase of a constant flow rate at the inner boundary the porousmedium is homogeneous isotropic and isothermal thesingle-phase horizontal flow does not have any gravity effectand the fluid and porous medium are slightly compressible

The continuous (mass balance) equation for the one-dimensional flow in the porous medium is given as follows[20 21 26 29]

1205972119875119863

1205971199092119863

=120597119875119863

120597119905119863

0 le 119909119863le 120575 (119905

119863) (1)

where 119875119863is the dimensionless formation pressure 119909

119863is the

dimensionless distance 119905119863is the dimensionless time and 120575 is

the dimensionless distance of the moving boundaryThe initial conditions are as follows

119875119863

1003816100381610038161003816119905119863=0= 0 (2)

120575 (0) = 0 (3)

The inner boundary conditionwith a constant productionrate is as follows

120597119875119863

120597119909119863

10038161003816100381610038161003816100381610038161003816119909119863=0

= minus (1 + 120582119863) (4)

where 120582119863is the dimensionless TPG

The moving boundary conditions are as follows [20 2126 29]

120597119875119863

120597119909119863

10038161003816100381610038161003816100381610038161003816119909119863=120575(119905119863)

= minus120582119863 (5)

119875119863

1003816100381610038161003816119909119863=120575(119905119863)= 0 (6)

Mathematical Problems in Engineering 3

Equations (1)ndash(6) form the mathematical model for theone-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary from (1) (5) and (6) the velocity of the movingboundary can be deduced as follows [29]

120597120575

120597119905119863

=1

120582119863

sdot1205972119875119863

1205971199092119863

100381610038161003816100381610038161003816100381610038161003816119909119863=120575(119905119863)

(7)

3 Numerical Solution of the Problem

It is well known that numerical solution needs the spacediscretization for the flow region however due to theexistence of the moving boundary in the model the one-dimensional flow region is not fixed but expands outsidecontinuously with the time increasing according to (7) Inorder to overcome the difficulty in the space discretizationfor the transient flow region with the moving boundary aspatial coordinate transformation method is introduced [33ndash35]Then themathematicalmodel with themoving boundaryconditions can be transformed into a mathematical modelwith fixed boundary conditions The formula of the spatialcoordinate transformation is as follows

119910119863(119909119863 119905119863) =

119909119863

120575 (119905119863) 0 le 119909

119863le 120575 (119905

119863) (8)

By the spatial coordinate transformation the followingtwo end points in the one-dimensional spatial coordinate of119909119863can be respectively transformed as

119910119863(0 119905119863) =

0

120575 (119905119863)= 0

119910119863(120575 (119905119863) 119905119863) =

120575 (119905119863)

120575 (119905119863)= 1

(9)

By (8) the dimensionless pressure 119875119863(119909119863 119905119863) at the flow

interval of the moving boundary model [0 120575(119905119863)] in the

one-dimensional spatial coordinate of 119909119863can be equivalently

transformed to the function 119863(119910119863 119905119863) at the fixed interval

[0 1] in the one-dimensional spatial coordinate of 119910119863 and

then the differential variables in the mathematical model canbe transformed respectively as follows

120597119875119863

120597119909119863

=120597119863

120597119910119863

sdot1

120575 (10)

1205972119875119863

1205971199092119863

=1205972119863

1205971199102119863

sdot1

1205752 (11)

120597119875119863

120597119905119863

=120597119863

120597119905119863

+120597119863

120597119910119863

sdot (minus119909119863

1205752) sdot

120597120575

120597119905119863

=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575

(12)

Substituting (10)ndash(12) into (1) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575 0 le 119910

119863le 1 (13)

Substituting (10)ndash(12) into (4)ndash(7) respectively yields

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=0

= minus (1 + 120582119863) (14)

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= minus120582119863 (15)

119863

10038161003816100381610038161003816119910119863=1= 0 (16)

120597120575

120597119905119863

=1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot1

1205752 (17)

Substituting (17) into (13) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot119910119863

1205753 (18)

that is

1205972119863

1205971199102119863

sdot 120575 =120597119863

120597119905119863

sdot 1205753minus120597119863

120597119910119863

sdot119910119863

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(19)

The equivalent form of (14) is as follows

120575 = minus120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

sdot1

1 + 120582119863

(20)

Substituting (20) into (19) to cancel the variable 120575 yields

1205972119863

1205971199102119863

sdot120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

minus1

(1 + 120582119863)2(120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

)

3

sdot120597119863

120597119905119863

minus1 + 120582119863

120582119863

sdot 119910119863sdot120597119863

120597119910119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= 0

(21)

From (8) (2) can be transformed as

119863(119910119863 119905119863)10038161003816100381610038161003816119905119863=0

= 0 (22)

From (14) and (15) the following equation can bededuced

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

=1 + 120582119863

120582119863

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(23)

Equations (21)ndash(23) and (16) form a closed system ofpartial differential equations with the fixed boundary condi-tions with respect to

119863(119910119863 119905119863) which is equivalent to the

mathematical model for the moving boundary problem ofthe one-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary From (21) it can be seen that the partial differentialequations have strong nonlinearity which indirectly reflects

4 Mathematical Problems in Engineering

the strong nonlinearity of the original moving boundarymodel Owing to the strong nonlinearity of the transformedmodel a stable fully implicit finite difference method [36] isadopted to numerically solve the relevant nonlinear partialdifferential equations

The one-dimensional unit interval is discretized as 119873

spatial grid subintervals with the same length and then thelength of every subinterval Δ119910

119863is equal to 1119873 the distance

at the 119894th spatial grid 119910119863119894

is 119894 sdot Δ119910119863 the time step is assumed

to be Δ119905119863 replace the first derivative by first-order forward

difference and the second derivative by second-order centraldifference and finally the fully implicit difference equations[36] corresponding to (21) can be expressed as follows

119895+1

119863119894+1minus 2119895+1

119863119894+ 119895+1

119863119894minus1

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119894minus 119895

119863119894

Δ119905119863

minus1 + 120582119863

120582119863

sdot 119894 sdot Δ119910119863sdot

119895+1

119863119894+1minus 119895+1

119863119894

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(119894 = 1 2 119873 minus 2)

(24)

where 119895 denotes the index of time stepThe difference equation of (16) is as follows

119895+1

119863119873= 0 (25)

Then by (25) the difference equation corresponding tothe (119873minus1)th spatial grid can be expressed as follows

minus2119895+1

119863119873minus1+ 119895+1

119863119873minus2

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119873minus1minus 119895

119863119873minus1

Δ119905119863

+1 + 120582119863

120582119863

sdot (119873 minus 1) sdot Δ119910119863sdot

119895+1

119863119873minus1

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(26)

The difference equation of (23) is as follows

119895+1

1198631minus 119895+1

1198630

Δ119910119863

= minus1 + 120582119863

120582119863

119895+1

119863119873minus1

Δ119910119863

(27)

From (22) the initial values of the unknown dimension-less formation pressure are as follows

0

119863119894= 0 119894 = 0 1 2 119873 minus 1 (28)

From (24) (26) and (27) it can be seen that there are119873difference equations at the (119895 + 1)th time step which contain

119873 unknown variables 119895+1119863119894

(119894 = 0 1 2 119873 minus 1) due to thestrong nonlinearity of these difference equations theNewton-Raphson iteration method [36] is adopted to obtain theirnumerical solutions when

119895+1

119863119894(119894 = 0 1 2 119873 minus 1) are

numerically solved for let 119895 be equal to 119895 + 1 and in the samemanner the numerical solutions of 119873 difference equationswith respect to119873 unknown variables 119895+2

119863119894(119894 = 0 1 2 119873minus

1) at the (119895 + 2)th time step can also be numerically solvedfor the rest can be deduced by analogy then the numericalsolutions of the nonlinear partial differential equations withrespect to

119863(119910119863 119905119863) can be obtained

The difference equation of (8) is

119909119895+1

119863119894

= 119910119863119894

sdot 120575119895+1

(29)

The difference equation of (20) is

120575119895+1

= minus

119895+1

1198631minus 119895+1

1198630

Δ119910119863

sdot1

1 + 120582119863

(30)

Substituting (30) into (29) yields

119909119895+1

119863119894

= minus119894 sdot

119895+1

1198631minus 119895+1

1198630

1 + 120582119863

(31)

From (31) in the process of numerical solutions at everytime step the one-dimensional spatial coordinate of 119910

119863can

be transformed as the one of 119909119863 and then the numerical

solutions of 119863(119910119863 119905119863) can be transformed as the numerical

solutions of 119875119863(119909119863 119905119863) From (31) it can also be seen that

it is just the spatial coordinate transformation that is (8)that lets the time-dependent space discretization in the spatialcoordinate of119909

119863be transformed as a time-independent space

discretization in the spatial coordinate of 119910119863 which makes

the numerical solutions by the finite difference method moreapplicable and simple

4 Verification of Numerical Solutions

41 Exact Analytical Solutions The exact analytical solutionof the mathematical model for the one-dimensional flow insemi-infinite long porous media with TPG for the case of aconstant flow rate at the inner boundary is as follows [29]

119875119863(119909119863 119905119863)

= 2 (1 + 120582119863)

times [120579(119905119863

12119890minus1199092

1198634119905119863+119909

119863212058712 erf (119909

1198632radic119905

119863)

119890minus1205792

+12058712120579 erf (120579))minus

119909119863

2]

119909119863isin [0 120575]

(32)

where 120579 can be determined by the following equation withrespect to the TPG [29]

119890minus1205792

119890minus1205792

+ 12058712120579 erf (120579)=

120582119863

1 + 120582119863

(33)

Mathematical Problems in Engineering 5

Table 1 Calculation of dimensionless TPG 120582119863through experimental data [2] of eight samples

Sample 120582

(psicm)119896120583

(mdcp)Minimum flow

rate 1199021(cm3min)

Maximum flowrate 119902

2(cm3min)

Cross-sectionalarea 119860 (cm2)

Average flow velocity]119908= (1199021+ 1199022)1198602

(cmmin)120582119863= 120582119896120583]

119908

Brown sandstone 1 000846 6520 883 30 202 0961139 0242Brown sandstone 2 0014 4810 1319 3232 202 1126485 0252Brown sandstone 3 00203 4730 171 175 202 0475495 0852Brown sandstone 4 0017 3130 579 2046 202 0649752 0345Sandpack 1 00046 27900 381 343 114 1671491 0324Sandpack 2 000789 23500 247 2034 114 1000439 0782Sandpack 3 000548 23800 457 2992 114 1512719 0364Shaly sandstone 0813 566 2 6 114 0350877 0553

It has been proven [29] that for a given value of 120582119863 there

is one and only one positive root of (33)The distance of the moving boundary can be expressed as

follows [29]

120575 = 2 sdot 120579 sdot radic119905119863 (34)

When 120582119863= 0 the exact analytical solution of the mathe-

matical model for the one-dimensional Darcyrsquos flow in semi-infinite long porous media for the case of a constant flow rateat the inner boundary is as follows [29]

119875119863(119909119863 119905119863) = minus 119909

119863+ 2radic

119905119863

120587exp(

minus1199092

119863

4119905119863

)

+ 119909119863erf ( 119909

119863

2radic119905119863

) 119909119863isin [0infin]

(35)

42 Assignment of the Values of Dimensionless TPG 120582119863 The

formula for the dimensionless TPG 120582119863is as follows [29]

120582119863=

119896

120583

120582

]119908

(36)

where 119896 is the permeability of porous media 120582 is the TPG 120583is the fluid viscosity and ]

119908is the average seepage velocity

Here selected Prada and Civanrsquo s actual experimentaldata of eight samples [2] for measuring the one-dimensionalflow in three types of porous media (brown sandstone sand-pack and shaly sandstone) with TPG is used to calculate thevalues of 120582

119863 the experimental data and specific calculation

process are shown in Table 1From Table 1 it can be concluded that the range of the

values of dimensionless TPG 120582119863is between 0242 and 0852

for the eight samples of three types of porous mediaWithoutloss of generality the values of dimensionless TPG 120582

119863are

set 0242 0553 and 0852 for the following numerical testsrespectively

43 Numerical Tests By the numerical method presented inthe paper the mathematical model for the one-dimensionalflow in semi-infinite long porousmedia with TPG for the caseof a constant flow rate at the inner boundary is numerically

00 20 40 60 80 10tD

200

160

120

80

40

0

120575(tD

)

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0852 0553 0242

103

Figure 1 Comparison of transient distance of moving boundaryunder different values of dimensionless TPG

solved where 119873 = 160 Δ119905119863= 10 and 120582

119863= 0242 0553 and

0852 respectively By (33) the values of 120579 correspondingto the three values of 120582

119863can be computed by Newton-

Raphson iteration method [36] as 1005 07732 and 06599for the exact analytical solutions In order to make clear thedifference of model solutions between Darcyrsquos flow when120582119863= 0 and the fluid flow in porous media with the TPG

exact analytical solutions for the Darcyrsquos flow that is (35) arealso added for the purpose of comparison analysis

Under different values of 120582119863 the numerical results are

graphically compared with those from the exact analyticalsolution that is (32)ndash(34) Figures 1ndash3 show the compari-son curves between the numerical solutions and the exactanalytical solutions with respect to the transient distance ofthe moving boundary the transient production pressure ofwellbore and the formation pressure distribution when 119905

119863=

5000 respectivelyFrom Figures 1ndash3 it can be seen that the numerical

solutions are in good agreement with the exact analytical

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Equations (1)ndash(6) form the mathematical model for theone-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary from (1) (5) and (6) the velocity of the movingboundary can be deduced as follows [29]

120597120575

120597119905119863

=1

120582119863

sdot1205972119875119863

1205971199092119863

100381610038161003816100381610038161003816100381610038161003816119909119863=120575(119905119863)

(7)

3 Numerical Solution of the Problem

It is well known that numerical solution needs the spacediscretization for the flow region however due to theexistence of the moving boundary in the model the one-dimensional flow region is not fixed but expands outsidecontinuously with the time increasing according to (7) Inorder to overcome the difficulty in the space discretizationfor the transient flow region with the moving boundary aspatial coordinate transformation method is introduced [33ndash35]Then themathematicalmodel with themoving boundaryconditions can be transformed into a mathematical modelwith fixed boundary conditions The formula of the spatialcoordinate transformation is as follows

119910119863(119909119863 119905119863) =

119909119863

120575 (119905119863) 0 le 119909

119863le 120575 (119905

119863) (8)

By the spatial coordinate transformation the followingtwo end points in the one-dimensional spatial coordinate of119909119863can be respectively transformed as

119910119863(0 119905119863) =

0

120575 (119905119863)= 0

119910119863(120575 (119905119863) 119905119863) =

120575 (119905119863)

120575 (119905119863)= 1

(9)

By (8) the dimensionless pressure 119875119863(119909119863 119905119863) at the flow

interval of the moving boundary model [0 120575(119905119863)] in the

one-dimensional spatial coordinate of 119909119863can be equivalently

transformed to the function 119863(119910119863 119905119863) at the fixed interval

[0 1] in the one-dimensional spatial coordinate of 119910119863 and

then the differential variables in the mathematical model canbe transformed respectively as follows

120597119875119863

120597119909119863

=120597119863

120597119910119863

sdot1

120575 (10)

1205972119875119863

1205971199092119863

=1205972119863

1205971199102119863

sdot1

1205752 (11)

120597119875119863

120597119905119863

=120597119863

120597119905119863

+120597119863

120597119910119863

sdot (minus119909119863

1205752) sdot

120597120575

120597119905119863

=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575

(12)

Substituting (10)ndash(12) into (1) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot120597120575

120597119905119863

sdot119910119863

120575 0 le 119910

119863le 1 (13)

Substituting (10)ndash(12) into (4)ndash(7) respectively yields

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=0

= minus (1 + 120582119863) (14)

120597119863

120597119910119863

sdot1

120575

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= minus120582119863 (15)

119863

10038161003816100381610038161003816119910119863=1= 0 (16)

120597120575

120597119905119863

=1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot1

1205752 (17)

Substituting (17) into (13) yields

1205972119863

1205971199102119863

sdot1

1205752=120597119863

120597119905119863

minus120597119863

120597119910119863

sdot1

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

sdot119910119863

1205753 (18)

that is

1205972119863

1205971199102119863

sdot 120575 =120597119863

120597119905119863

sdot 1205753minus120597119863

120597119910119863

sdot119910119863

120582119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(19)

The equivalent form of (14) is as follows

120575 = minus120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

sdot1

1 + 120582119863

(20)

Substituting (20) into (19) to cancel the variable 120575 yields

1205972119863

1205971199102119863

sdot120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

minus1

(1 + 120582119863)2(120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

)

3

sdot120597119863

120597119905119863

minus1 + 120582119863

120582119863

sdot 119910119863sdot120597119863

120597119910119863

sdot1205972119863

1205971199102119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

= 0

(21)

From (8) (2) can be transformed as

119863(119910119863 119905119863)10038161003816100381610038161003816119905119863=0

= 0 (22)

From (14) and (15) the following equation can bededuced

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=0

=1 + 120582119863

120582119863

120597119863

120597119910119863

100381610038161003816100381610038161003816100381610038161003816119910119863=1

(23)

Equations (21)ndash(23) and (16) form a closed system ofpartial differential equations with the fixed boundary condi-tions with respect to

119863(119910119863 119905119863) which is equivalent to the

mathematical model for the moving boundary problem ofthe one-dimensional flow in semi-infinite long porous mediawith TPG for the case of a constant flow rate at the innerboundary From (21) it can be seen that the partial differentialequations have strong nonlinearity which indirectly reflects

4 Mathematical Problems in Engineering

the strong nonlinearity of the original moving boundarymodel Owing to the strong nonlinearity of the transformedmodel a stable fully implicit finite difference method [36] isadopted to numerically solve the relevant nonlinear partialdifferential equations

The one-dimensional unit interval is discretized as 119873

spatial grid subintervals with the same length and then thelength of every subinterval Δ119910

119863is equal to 1119873 the distance

at the 119894th spatial grid 119910119863119894

is 119894 sdot Δ119910119863 the time step is assumed

to be Δ119905119863 replace the first derivative by first-order forward

difference and the second derivative by second-order centraldifference and finally the fully implicit difference equations[36] corresponding to (21) can be expressed as follows

119895+1

119863119894+1minus 2119895+1

119863119894+ 119895+1

119863119894minus1

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119894minus 119895

119863119894

Δ119905119863

minus1 + 120582119863

120582119863

sdot 119894 sdot Δ119910119863sdot

119895+1

119863119894+1minus 119895+1

119863119894

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(119894 = 1 2 119873 minus 2)

(24)

where 119895 denotes the index of time stepThe difference equation of (16) is as follows

119895+1

119863119873= 0 (25)

Then by (25) the difference equation corresponding tothe (119873minus1)th spatial grid can be expressed as follows

minus2119895+1

119863119873minus1+ 119895+1

119863119873minus2

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119873minus1minus 119895

119863119873minus1

Δ119905119863

+1 + 120582119863

120582119863

sdot (119873 minus 1) sdot Δ119910119863sdot

119895+1

119863119873minus1

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(26)

The difference equation of (23) is as follows

119895+1

1198631minus 119895+1

1198630

Δ119910119863

= minus1 + 120582119863

120582119863

119895+1

119863119873minus1

Δ119910119863

(27)

From (22) the initial values of the unknown dimension-less formation pressure are as follows

0

119863119894= 0 119894 = 0 1 2 119873 minus 1 (28)

From (24) (26) and (27) it can be seen that there are119873difference equations at the (119895 + 1)th time step which contain

119873 unknown variables 119895+1119863119894

(119894 = 0 1 2 119873 minus 1) due to thestrong nonlinearity of these difference equations theNewton-Raphson iteration method [36] is adopted to obtain theirnumerical solutions when

119895+1

119863119894(119894 = 0 1 2 119873 minus 1) are

numerically solved for let 119895 be equal to 119895 + 1 and in the samemanner the numerical solutions of 119873 difference equationswith respect to119873 unknown variables 119895+2

119863119894(119894 = 0 1 2 119873minus

1) at the (119895 + 2)th time step can also be numerically solvedfor the rest can be deduced by analogy then the numericalsolutions of the nonlinear partial differential equations withrespect to

119863(119910119863 119905119863) can be obtained

The difference equation of (8) is

119909119895+1

119863119894

= 119910119863119894

sdot 120575119895+1

(29)

The difference equation of (20) is

120575119895+1

= minus

119895+1

1198631minus 119895+1

1198630

Δ119910119863

sdot1

1 + 120582119863

(30)

Substituting (30) into (29) yields

119909119895+1

119863119894

= minus119894 sdot

119895+1

1198631minus 119895+1

1198630

1 + 120582119863

(31)

From (31) in the process of numerical solutions at everytime step the one-dimensional spatial coordinate of 119910

119863can

be transformed as the one of 119909119863 and then the numerical

solutions of 119863(119910119863 119905119863) can be transformed as the numerical

solutions of 119875119863(119909119863 119905119863) From (31) it can also be seen that

it is just the spatial coordinate transformation that is (8)that lets the time-dependent space discretization in the spatialcoordinate of119909

119863be transformed as a time-independent space

discretization in the spatial coordinate of 119910119863 which makes

the numerical solutions by the finite difference method moreapplicable and simple

4 Verification of Numerical Solutions

41 Exact Analytical Solutions The exact analytical solutionof the mathematical model for the one-dimensional flow insemi-infinite long porous media with TPG for the case of aconstant flow rate at the inner boundary is as follows [29]

119875119863(119909119863 119905119863)

= 2 (1 + 120582119863)

times [120579(119905119863

12119890minus1199092

1198634119905119863+119909

119863212058712 erf (119909

1198632radic119905

119863)

119890minus1205792

+12058712120579 erf (120579))minus

119909119863

2]

119909119863isin [0 120575]

(32)

where 120579 can be determined by the following equation withrespect to the TPG [29]

119890minus1205792

119890minus1205792

+ 12058712120579 erf (120579)=

120582119863

1 + 120582119863

(33)

Mathematical Problems in Engineering 5

Table 1 Calculation of dimensionless TPG 120582119863through experimental data [2] of eight samples

Sample 120582

(psicm)119896120583

(mdcp)Minimum flow

rate 1199021(cm3min)

Maximum flowrate 119902

2(cm3min)

Cross-sectionalarea 119860 (cm2)

Average flow velocity]119908= (1199021+ 1199022)1198602

(cmmin)120582119863= 120582119896120583]

119908

Brown sandstone 1 000846 6520 883 30 202 0961139 0242Brown sandstone 2 0014 4810 1319 3232 202 1126485 0252Brown sandstone 3 00203 4730 171 175 202 0475495 0852Brown sandstone 4 0017 3130 579 2046 202 0649752 0345Sandpack 1 00046 27900 381 343 114 1671491 0324Sandpack 2 000789 23500 247 2034 114 1000439 0782Sandpack 3 000548 23800 457 2992 114 1512719 0364Shaly sandstone 0813 566 2 6 114 0350877 0553

It has been proven [29] that for a given value of 120582119863 there

is one and only one positive root of (33)The distance of the moving boundary can be expressed as

follows [29]

120575 = 2 sdot 120579 sdot radic119905119863 (34)

When 120582119863= 0 the exact analytical solution of the mathe-

matical model for the one-dimensional Darcyrsquos flow in semi-infinite long porous media for the case of a constant flow rateat the inner boundary is as follows [29]

119875119863(119909119863 119905119863) = minus 119909

119863+ 2radic

119905119863

120587exp(

minus1199092

119863

4119905119863

)

+ 119909119863erf ( 119909

119863

2radic119905119863

) 119909119863isin [0infin]

(35)

42 Assignment of the Values of Dimensionless TPG 120582119863 The

formula for the dimensionless TPG 120582119863is as follows [29]

120582119863=

119896

120583

120582

]119908

(36)

where 119896 is the permeability of porous media 120582 is the TPG 120583is the fluid viscosity and ]

119908is the average seepage velocity

Here selected Prada and Civanrsquo s actual experimentaldata of eight samples [2] for measuring the one-dimensionalflow in three types of porous media (brown sandstone sand-pack and shaly sandstone) with TPG is used to calculate thevalues of 120582

119863 the experimental data and specific calculation

process are shown in Table 1From Table 1 it can be concluded that the range of the

values of dimensionless TPG 120582119863is between 0242 and 0852

for the eight samples of three types of porous mediaWithoutloss of generality the values of dimensionless TPG 120582

119863are

set 0242 0553 and 0852 for the following numerical testsrespectively

43 Numerical Tests By the numerical method presented inthe paper the mathematical model for the one-dimensionalflow in semi-infinite long porousmedia with TPG for the caseof a constant flow rate at the inner boundary is numerically

00 20 40 60 80 10tD

200

160

120

80

40

0

120575(tD

)

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0852 0553 0242

103

Figure 1 Comparison of transient distance of moving boundaryunder different values of dimensionless TPG

solved where 119873 = 160 Δ119905119863= 10 and 120582

119863= 0242 0553 and

0852 respectively By (33) the values of 120579 correspondingto the three values of 120582

119863can be computed by Newton-

Raphson iteration method [36] as 1005 07732 and 06599for the exact analytical solutions In order to make clear thedifference of model solutions between Darcyrsquos flow when120582119863= 0 and the fluid flow in porous media with the TPG

exact analytical solutions for the Darcyrsquos flow that is (35) arealso added for the purpose of comparison analysis

Under different values of 120582119863 the numerical results are

graphically compared with those from the exact analyticalsolution that is (32)ndash(34) Figures 1ndash3 show the compari-son curves between the numerical solutions and the exactanalytical solutions with respect to the transient distance ofthe moving boundary the transient production pressure ofwellbore and the formation pressure distribution when 119905

119863=

5000 respectivelyFrom Figures 1ndash3 it can be seen that the numerical

solutions are in good agreement with the exact analytical

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

the strong nonlinearity of the original moving boundarymodel Owing to the strong nonlinearity of the transformedmodel a stable fully implicit finite difference method [36] isadopted to numerically solve the relevant nonlinear partialdifferential equations

The one-dimensional unit interval is discretized as 119873

spatial grid subintervals with the same length and then thelength of every subinterval Δ119910

119863is equal to 1119873 the distance

at the 119894th spatial grid 119910119863119894

is 119894 sdot Δ119910119863 the time step is assumed

to be Δ119905119863 replace the first derivative by first-order forward

difference and the second derivative by second-order centraldifference and finally the fully implicit difference equations[36] corresponding to (21) can be expressed as follows

119895+1

119863119894+1minus 2119895+1

119863119894+ 119895+1

119863119894minus1

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119894minus 119895

119863119894

Δ119905119863

minus1 + 120582119863

120582119863

sdot 119894 sdot Δ119910119863sdot

119895+1

119863119894+1minus 119895+1

119863119894

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(119894 = 1 2 119873 minus 2)

(24)

where 119895 denotes the index of time stepThe difference equation of (16) is as follows

119895+1

119863119873= 0 (25)

Then by (25) the difference equation corresponding tothe (119873minus1)th spatial grid can be expressed as follows

minus2119895+1

119863119873minus1+ 119895+1

119863119873minus2

(Δ119910119863)2

sdot

119895+1

1198631minus 119895+1

1198630

Δ119910119863

minus1

(1 + 120582119863)2(

119895+1

1198631minus 119895+1

1198630

Δ119910119863

)

3

sdot

119895+1

119863119873minus1minus 119895

119863119873minus1

Δ119905119863

+1 + 120582119863

120582119863

sdot (119873 minus 1) sdot Δ119910119863sdot

119895+1

119863119873minus1

Δ119910119863

sdot

119895+1

119863119873minus2minus 2119895+1

119863119873minus1

(Δ119910119863)2

= 0

(26)

The difference equation of (23) is as follows

119895+1

1198631minus 119895+1

1198630

Δ119910119863

= minus1 + 120582119863

120582119863

119895+1

119863119873minus1

Δ119910119863

(27)

From (22) the initial values of the unknown dimension-less formation pressure are as follows

0

119863119894= 0 119894 = 0 1 2 119873 minus 1 (28)

From (24) (26) and (27) it can be seen that there are119873difference equations at the (119895 + 1)th time step which contain

119873 unknown variables 119895+1119863119894

(119894 = 0 1 2 119873 minus 1) due to thestrong nonlinearity of these difference equations theNewton-Raphson iteration method [36] is adopted to obtain theirnumerical solutions when

119895+1

119863119894(119894 = 0 1 2 119873 minus 1) are

numerically solved for let 119895 be equal to 119895 + 1 and in the samemanner the numerical solutions of 119873 difference equationswith respect to119873 unknown variables 119895+2

119863119894(119894 = 0 1 2 119873minus

1) at the (119895 + 2)th time step can also be numerically solvedfor the rest can be deduced by analogy then the numericalsolutions of the nonlinear partial differential equations withrespect to

119863(119910119863 119905119863) can be obtained

The difference equation of (8) is

119909119895+1

119863119894

= 119910119863119894

sdot 120575119895+1

(29)

The difference equation of (20) is

120575119895+1

= minus

119895+1

1198631minus 119895+1

1198630

Δ119910119863

sdot1

1 + 120582119863

(30)

Substituting (30) into (29) yields

119909119895+1

119863119894

= minus119894 sdot

119895+1

1198631minus 119895+1

1198630

1 + 120582119863

(31)

From (31) in the process of numerical solutions at everytime step the one-dimensional spatial coordinate of 119910

119863can

be transformed as the one of 119909119863 and then the numerical

solutions of 119863(119910119863 119905119863) can be transformed as the numerical

solutions of 119875119863(119909119863 119905119863) From (31) it can also be seen that

it is just the spatial coordinate transformation that is (8)that lets the time-dependent space discretization in the spatialcoordinate of119909

119863be transformed as a time-independent space

discretization in the spatial coordinate of 119910119863 which makes

the numerical solutions by the finite difference method moreapplicable and simple

4 Verification of Numerical Solutions

41 Exact Analytical Solutions The exact analytical solutionof the mathematical model for the one-dimensional flow insemi-infinite long porous media with TPG for the case of aconstant flow rate at the inner boundary is as follows [29]

119875119863(119909119863 119905119863)

= 2 (1 + 120582119863)

times [120579(119905119863

12119890minus1199092

1198634119905119863+119909

119863212058712 erf (119909

1198632radic119905

119863)

119890minus1205792

+12058712120579 erf (120579))minus

119909119863

2]

119909119863isin [0 120575]

(32)

where 120579 can be determined by the following equation withrespect to the TPG [29]

119890minus1205792

119890minus1205792

+ 12058712120579 erf (120579)=

120582119863

1 + 120582119863

(33)

Mathematical Problems in Engineering 5

Table 1 Calculation of dimensionless TPG 120582119863through experimental data [2] of eight samples

Sample 120582

(psicm)119896120583

(mdcp)Minimum flow

rate 1199021(cm3min)

Maximum flowrate 119902

2(cm3min)

Cross-sectionalarea 119860 (cm2)

Average flow velocity]119908= (1199021+ 1199022)1198602

(cmmin)120582119863= 120582119896120583]

119908

Brown sandstone 1 000846 6520 883 30 202 0961139 0242Brown sandstone 2 0014 4810 1319 3232 202 1126485 0252Brown sandstone 3 00203 4730 171 175 202 0475495 0852Brown sandstone 4 0017 3130 579 2046 202 0649752 0345Sandpack 1 00046 27900 381 343 114 1671491 0324Sandpack 2 000789 23500 247 2034 114 1000439 0782Sandpack 3 000548 23800 457 2992 114 1512719 0364Shaly sandstone 0813 566 2 6 114 0350877 0553

It has been proven [29] that for a given value of 120582119863 there

is one and only one positive root of (33)The distance of the moving boundary can be expressed as

follows [29]

120575 = 2 sdot 120579 sdot radic119905119863 (34)

When 120582119863= 0 the exact analytical solution of the mathe-

matical model for the one-dimensional Darcyrsquos flow in semi-infinite long porous media for the case of a constant flow rateat the inner boundary is as follows [29]

119875119863(119909119863 119905119863) = minus 119909

119863+ 2radic

119905119863

120587exp(

minus1199092

119863

4119905119863

)

+ 119909119863erf ( 119909

119863

2radic119905119863

) 119909119863isin [0infin]

(35)

42 Assignment of the Values of Dimensionless TPG 120582119863 The

formula for the dimensionless TPG 120582119863is as follows [29]

120582119863=

119896

120583

120582

]119908

(36)

where 119896 is the permeability of porous media 120582 is the TPG 120583is the fluid viscosity and ]

119908is the average seepage velocity

Here selected Prada and Civanrsquo s actual experimentaldata of eight samples [2] for measuring the one-dimensionalflow in three types of porous media (brown sandstone sand-pack and shaly sandstone) with TPG is used to calculate thevalues of 120582

119863 the experimental data and specific calculation

process are shown in Table 1From Table 1 it can be concluded that the range of the

values of dimensionless TPG 120582119863is between 0242 and 0852

for the eight samples of three types of porous mediaWithoutloss of generality the values of dimensionless TPG 120582

119863are

set 0242 0553 and 0852 for the following numerical testsrespectively

43 Numerical Tests By the numerical method presented inthe paper the mathematical model for the one-dimensionalflow in semi-infinite long porousmedia with TPG for the caseof a constant flow rate at the inner boundary is numerically

00 20 40 60 80 10tD

200

160

120

80

40

0

120575(tD

)

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0852 0553 0242

103

Figure 1 Comparison of transient distance of moving boundaryunder different values of dimensionless TPG

solved where 119873 = 160 Δ119905119863= 10 and 120582

119863= 0242 0553 and

0852 respectively By (33) the values of 120579 correspondingto the three values of 120582

119863can be computed by Newton-

Raphson iteration method [36] as 1005 07732 and 06599for the exact analytical solutions In order to make clear thedifference of model solutions between Darcyrsquos flow when120582119863= 0 and the fluid flow in porous media with the TPG

exact analytical solutions for the Darcyrsquos flow that is (35) arealso added for the purpose of comparison analysis

Under different values of 120582119863 the numerical results are

graphically compared with those from the exact analyticalsolution that is (32)ndash(34) Figures 1ndash3 show the compari-son curves between the numerical solutions and the exactanalytical solutions with respect to the transient distance ofthe moving boundary the transient production pressure ofwellbore and the formation pressure distribution when 119905

119863=

5000 respectivelyFrom Figures 1ndash3 it can be seen that the numerical

solutions are in good agreement with the exact analytical

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Calculation of dimensionless TPG 120582119863through experimental data [2] of eight samples

Sample 120582

(psicm)119896120583

(mdcp)Minimum flow

rate 1199021(cm3min)

Maximum flowrate 119902

2(cm3min)

Cross-sectionalarea 119860 (cm2)

Average flow velocity]119908= (1199021+ 1199022)1198602

(cmmin)120582119863= 120582119896120583]

119908

Brown sandstone 1 000846 6520 883 30 202 0961139 0242Brown sandstone 2 0014 4810 1319 3232 202 1126485 0252Brown sandstone 3 00203 4730 171 175 202 0475495 0852Brown sandstone 4 0017 3130 579 2046 202 0649752 0345Sandpack 1 00046 27900 381 343 114 1671491 0324Sandpack 2 000789 23500 247 2034 114 1000439 0782Sandpack 3 000548 23800 457 2992 114 1512719 0364Shaly sandstone 0813 566 2 6 114 0350877 0553

It has been proven [29] that for a given value of 120582119863 there

is one and only one positive root of (33)The distance of the moving boundary can be expressed as

follows [29]

120575 = 2 sdot 120579 sdot radic119905119863 (34)

When 120582119863= 0 the exact analytical solution of the mathe-

matical model for the one-dimensional Darcyrsquos flow in semi-infinite long porous media for the case of a constant flow rateat the inner boundary is as follows [29]

119875119863(119909119863 119905119863) = minus 119909

119863+ 2radic

119905119863

120587exp(

minus1199092

119863

4119905119863

)

+ 119909119863erf ( 119909

119863

2radic119905119863

) 119909119863isin [0infin]

(35)

42 Assignment of the Values of Dimensionless TPG 120582119863 The

formula for the dimensionless TPG 120582119863is as follows [29]

120582119863=

119896

120583

120582

]119908

(36)

where 119896 is the permeability of porous media 120582 is the TPG 120583is the fluid viscosity and ]

119908is the average seepage velocity

Here selected Prada and Civanrsquo s actual experimentaldata of eight samples [2] for measuring the one-dimensionalflow in three types of porous media (brown sandstone sand-pack and shaly sandstone) with TPG is used to calculate thevalues of 120582

119863 the experimental data and specific calculation

process are shown in Table 1From Table 1 it can be concluded that the range of the

values of dimensionless TPG 120582119863is between 0242 and 0852

for the eight samples of three types of porous mediaWithoutloss of generality the values of dimensionless TPG 120582

119863are

set 0242 0553 and 0852 for the following numerical testsrespectively

43 Numerical Tests By the numerical method presented inthe paper the mathematical model for the one-dimensionalflow in semi-infinite long porousmedia with TPG for the caseof a constant flow rate at the inner boundary is numerically

00 20 40 60 80 10tD

200

160

120

80

40

0

120575(tD

)

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0852 0553 0242

103

Figure 1 Comparison of transient distance of moving boundaryunder different values of dimensionless TPG

solved where 119873 = 160 Δ119905119863= 10 and 120582

119863= 0242 0553 and

0852 respectively By (33) the values of 120579 correspondingto the three values of 120582

119863can be computed by Newton-

Raphson iteration method [36] as 1005 07732 and 06599for the exact analytical solutions In order to make clear thedifference of model solutions between Darcyrsquos flow when120582119863= 0 and the fluid flow in porous media with the TPG

exact analytical solutions for the Darcyrsquos flow that is (35) arealso added for the purpose of comparison analysis

Under different values of 120582119863 the numerical results are

graphically compared with those from the exact analyticalsolution that is (32)ndash(34) Figures 1ndash3 show the compari-son curves between the numerical solutions and the exactanalytical solutions with respect to the transient distance ofthe moving boundary the transient production pressure ofwellbore and the formation pressure distribution when 119905

119863=

5000 respectivelyFrom Figures 1ndash3 it can be seen that the numerical

solutions are in good agreement with the exact analytical

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

20 40 60 8010

3

10tD

Analytical solutionNumerical solution

N = 160 Δt = 10

120582D = 0 0242 0553 0852

180

150

120

90

60

30

0

PD

(0t

D)

Figure 2 Comparison of transient production pressure of wellboreunder different values of dimensionless TPG

120582D = 0852 0553 0242 0

125

100

75

50

25

0

Analytical solutionNumerical solution

0 50 100 150 200 250

N = 160 Δt = 10

xD

PD(x

D5000)

Figure 3 Comparison of formation pressure distribution when 119905119863

= 5000 under different values of dimensionless TPG

solutions for the moving boundary problem Furthermorethrough lots of numerical tests it is found that the accuracyof the numerical solutions can be further improved byincreasing the value of119873 It demonstrates the correctness andvalidity of the presented numerical method here for solvingthe moving boundary problem of the one-dimensional flowin semi-infinite long porous media with TPG

Besides from Figures 2 and 3 it can also be concludedthat the TPG has a big influence on the model solutionsthe larger the value of the dimensionless TPG the largerthe difference of model solutions between Darcyrsquos flow andfluid flow in porous media with TPG in Figure 3 the forma-tion pressure distribution curves corresponding to non-zero

dimensionless TPG indicate an instructive characteristicsof model solutions they have compact supports [1] whichare different from that of Darcyrsquos flow Therefore it is verynecessary to take into account the effect of TPG for the fluidflow in porous media with TPG for mathematical modelingand engineering applications especially for the case of a largedimensionless TPG that is (36)

5 Conclusions

The utility of the presented numerical method can beattributed to its simple approach of spatial coordinate trans-formation in numerically solving themoving boundary prob-lem effectively by the stable fully implicit finite differencemethod The accuracy of the numerical solutions is verifiedby graphically comparing with the published exact analyticalsolutions Besides the accuracy can be further improved byincreasing the value of 119873 that is decreasing the length ofthe spatial grid subintervals The numerical results also showthe big difference of model solutions between Darcyrsquos flowand the fluid flow in porous media with TPG especially forthe case of a large dimensionless TPG In comparison withprevious numericalmethods for solving suchmoving bound-ary problems the verified numerical method presented hereis more reliable and convenient to implement by the finitedifference method in developing well testing software andreservoir simulatorThe numerical method will be applicableto the moving boundary problems of multi-dimensionalflow in porous media with TPG which will be our futureresearch topic The presented research supports theoreticalfoundations for technologies of well testing and numeri-cal simulation in developing low-permeable reservoirs andheavy oil reservoirs

Conflict of Interests

Jun YaoWenchao Liu and Zhangxin Chen declare that thereis no conflict of interests regarding the publication of thispaper

Acknowledgments

The authors would like to acknowledge the funding bythe project (Grant no 11102237) sponsored by the NaturalScience Foundation of China (NSFC) and the Programfor Changjiang Scholars and Innovative Research Team inUniversity (Grant no IRT1294) In particular Wenchao Liualso would like to express his great gratitude to the ChinaScholarship Council (CSC) for the generous financial supportof the research

References

[1] P J M Monteiro C H Rycroft and G I Barenblatt ldquoAmathematical model of fluid and gas flow in nanoporousmediardquo Proceedings of the National Academy of Sciences of theUnited States of America vol 109 no 50 pp 20309ndash20313 2012

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

[2] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[3] F Civan Porous Media Transport Phenomena John Wiley andSons Hoboken NJ USA 2011

[4] SWang Y Huang and F Civan ldquoExperimental and theoreticalinvestigation of the Zaoyuan field heavy oil flow through porousmediardquo Journal of Petroleum Science and Engineering vol 50no 2 pp 83ndash101 2006

[5] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[6] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[7] W Jing L Huiqing P Zhanxi L Renjing and L Ming ldquoTheinvestigation of threshold pressure gradient of foam flooding inporous mediardquo Petroleum Science and Technology vol 29 no23 pp 2460ndash2470 2011

[8] Y D Yao and J L Ge ldquoCharacteristics of non-Darcy flow inlow-permeability reservoirsrdquo Petroleum Science vol 8 no 1 pp55ndash62 2011

[9] F-Q Song J-DWang andH-L Liu ldquoStatic threshold pressuregradient characteristics of liquid influenced by boundary wetta-bilityrdquo Chinese Physics Letters vol 27 no 2 Article ID 0247042010

[10] X-W Wang Z-M Yang Y-D Qi and Y-Z Huang ldquoEffectof absorption boundary layer on nonlinear flow in low per-meability porous mediardquo Journal of Central South University ofTechnology vol 18 no 4 pp 1299ndash1303 2011

[11] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2012

[12] M Yun B Yu and J Cai ldquoA fractal model for the starting pres-sure gradient for Binghamfluids in porousmediardquo InternationalJournal ofHeat andMass Transfer vol 51 no 5-6 pp 1402ndash14082008

[13] Y Li and B Yu ldquoStudy of the starting pressure gradient inbranching networkrdquo Science China Technological Sciences vol53 no 9 pp 2397ndash2403 2010

[14] J C Cai B M Yu M Q Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[15] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energyand Fuels vol 24 no 3 pp 1860ndash1867 2010

[16] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energyand Fuels vol 25 no 3 pp 1111ndash1117 2011

[17] Y Zhu J-Z Xie W-H Yang and L-H Hou ldquoMethod forimproving history matching precision of reservoir numericalsimulationrdquo Petroleum Exploration and Development vol 35no 2 pp 225ndash229 2008

[18] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[19] H Pascal ldquoNonsteady flow through porous media in thepresence of a threshold gradientrdquo Acta Mechanica vol 39 no3-4 pp 207ndash224 1981

[20] Y S Wu K Pruess and P AWitherspoon ldquoFlow and displace-ment of Bingham non-Newtonian fluids in porous mediardquo SPEReservoir Engineering vol 7 no 3 pp 369ndash376 1992

[21] S Fuquan L Ciqun and L Fanhua ldquoTransient pressure of per-colation through one dimension porous media with thresholdpressure gradientrdquoAppliedMathematics andMechanics vol 20no 1 pp 27ndash35 1999

[22] J Nedoma ldquoNumerical solution of a Stefan-like problem inBingham rheologyrdquoMathematics and Computers in Simulationvol 61 no 3ndash6 pp 271ndash281 2003

[23] M Chen W Rossen and Y C Yortsos ldquoThe flow and dis-placement in porous media of fluids with yield stressrdquoChemicalEngineering Science vol 60 no 15 pp 4183ndash4202 2005

[24] G-Q Feng Q-G Liu G-Z Shi and Z-H Lin ldquoAn unsteadyseepage flow model considering kickoff pressure gradient forlow-permeability gas reservoirsrdquo Petroleum Exploration andDevelopment vol 35 no 4 pp 457ndash461 2008

[25] I Dapra and G Scarpi ldquoUnsteady simple shear flow in a vis-coplastic fluid comparison between analytical and numericalsolutionsrdquo Rheologica Acta vol 49 no 1 pp 15ndash22 2010

[26] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[27] W J Luo and X D Wang ldquoEffect of a moving boundary on thefluid transient flow in low permeability reservoirsrdquo Journal ofHydrodynamics B vol 24 no 3 pp 391ndash398 2012

[28] X Wang X Hou M Hao and T Yang ldquoPressure transientanalysis in low-permeable media with threshold gradientsrdquoActa Petrolei Sinica vol 32 no 5 pp 847ndash851 2011

[29] W C Liu J Yao and Y Y Wang ldquoExact analytical solutions ofmoving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradientrdquoInternational Journal of Heat Mass Transfer vol 55 no 21-22pp 6017ndash6022 2012

[30] W C Liu J Yao Z X Sun et al ldquoModel of nonlinearseepage flow in low-permeability porous media based on thepermeability gradual theoryrdquo Chinese Journal of ComputationalMechanics vol 29 no 6 pp 885ndash892 2012

[31] S L Marshall ldquoNonlinear pressure diffusion in flow of com-pressible liquids through porous mediardquo Transport in PorousMedia vol 77 no 3 pp 431ndash446 2009

[32] R Wang X Yue R Zhao P Yan and F Dave ldquoEffect ofstress sensitivity on displacement efficiency in CO

2flooding for

fractured low permeability reservoirsrdquo Petroleum Science vol 6no 3 pp 277ndash283 2009

[33] J Crank Free andMoving Boundary Problems Clarendon PressOxford UK 1984

[34] R S Gupta and A Kumar ldquoTreatment of multi-dimensionalmoving boundary problems by coordinate transformationrdquoInternational Journal of Heat and Mass Transfer vol 28 no 7pp 1355ndash1366 1985

[35] G Senthil G Jayaraman and A D Rao ldquoA variable boundarymethod for modelling two dimensional free surface flows withmoving boundariesrdquo Applied Mathematics and Computationvol 216 no 9 pp 2544ndash2558 2010

[36] R L Burden and J D Faires Numerical Analysis Brooks ColeFlorence Ky USA 9th edition 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of