research article a new mathematical model for pressure...

Research ArticleA New Mathematical Model for Pressure TransientAnalysis in Stress-Sensitive Reservoirs

Junjie Ren and Ping Guo

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityXindu Avenue No 8 Xindu District Chengdu City Sichuan 610500 China

Correspondence should be addressed to Junjie Ren renjunjie1900126com

Received 16 December 2013 Accepted 10 April 2014 Published 29 April 2014

Academic Editor Oluwole Daniel Makinde

Copyright copy 2014 J Ren and P Guo This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For stress-sensitive reservoir the permeability decreases with the increase of the effective overburden pressure and pressuretransient analysis based on constant rock properties especially permeability can lead to significant errors In this paper accordingto the permeability-pressure relationship described by a power function instead of the popularly used exponential function anew mathematical model for transient fluid flow in stress-sensitive reservoirs is established The numerical solution is obtainedby the fully implicit finite difference method which has been validated by some published analytical solutions before it is used tocompute pressure transient responses for stress-sensitive reservoirs Pressure response curves are plotted and the effects of relevantparameters on both pressure drawdown and buildup responses have been studiedThe presentedmodel could provide an alternativemethod for understanding and predicting the performances for stress-sensitive reservoirs

1 Introduction

In practice the effective overburden pressure of the reservoirwill increase with the decrease of the fluid pressure [1]Because the porous media are not always rigid and nonde-formable the increase of the effective overburden pressuremay result in plastic deformation or elastic deformation ofthe reservoir rocks and reduce the permeability porositycompression coefficient and so forth [2 3] To a certainextent the property of rock is pressure dependent or stresssensitive [4]

The production capacity of oil and gas wells is mainlyaffected by the permeability in reservoirs [5] In the devel-opment of oilfield the fluid of the reservoir is exploitedconstantly and the pore pressure in the reservoir decreasesgradually As a result it could cause the effective overburdenpressure to increase and lead to the compression and defor-mation of the rocks in the reservoir and a closure of the poresin the rocks Thus it may cause the permeability to reduceThe changes of permeability inevitably cause the changes ofpermeable capacity and then production capacity in oil wellis affected [6] Therefore it has been observed that measured

flow rates in stress-sensitive reservoirs are sometimes muchlower than the predicted ones by transient flow equationsbased on the assumption of constant rock properties [7 8]

Since the 1950s the laboratory studies and the math-ematical modeling research for stress-sensitive reservoirshave been reported by many investigators Laboratory exper-iments have found that the permeability decreases whenthe effective overburden pressure increases and this phe-nomenon is even more severe in tight samples [9ndash13]

In order to describe the transient flow behavior in stress-sensitive porous media several models have been presentedThe existing strategies for constructing the stress-sensitivemodels can be classified into two categories that is the pseu-dopressure approach and the permeability-stress functionapproach The pseudopressure approach has been applied tostudy pressure drawdown pressure buildup injection andfalloff testing by several investigators [14ndash16] The main dis-advantage of the pseudopressure approachwhichmayhinderits applications in practice is that the tabulated properties ofrock and fluids versus pressure should be known a priori ateach pressure level In addition a nonlinear diffusivity termstill exists in the diffusivity equation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 485028 14 pageshttpdxdoiorg1011552014485028

2 Mathematical Problems in Engineering

The permeability-stress function approach consideringthe pressure-dependent permeability as a permeability-stressfunction (eg exponential function) is widely used to studythe transient flow behavior in porous media through solvinga nonlinear diffusivity equation Based on the exponentialfunction relationship between the permeability and the porepressure (or the effective overburden pressure) Pedrosa [17]presented the first-order approximate analytical solution fora line-source well producing at a constant rate from aninfinite radial flow system using the perturbation techniqueKikani and Pedrosa [18] extended the work of Pedrosa [17] bypresenting the second-order approximate analytical solutionfor the same problem using the same solution method-ology and suggested the use of zero-order perturbationsolution to investigate the effects of wellbore storage skinand outer boundary on the pressure transient response forstress-sensitive reservoirs Zhang and Ambastha [19] furtherinvestigated the effects of stress-sensitive permeability ondrawdown and buildup pressure transient behavior using thestepwise permeability model and found that the pressuretransient response obtained by Kikani and Pedrosa [18] wasnot accurate when wellbore storage and skin were takeninto consideration so the numerical method was suggestedto study the pressure transient response in stress-sensitivereservoirs Ambastha and Zhang [20] extended the workof Zhang and Ambastha [19] by presenting three models(ie one-parametermodel stepwise permeabilitymodel andtwo-parameter model) for stress-sensitive reservoirs basedon the exponential function (or the modified exponentialfunction) relationship between the permeability and the porepressure (or the effective overburden pressure) Jelmert andSelseng [21] presented an exponential permeability modelthrough introducing normalized permeability variables andobtained approximate analytical solutions which are compu-tationally simple and readily available Wu and Pruess [22]presented an integral method for describing the transientflow behavior in stress-sensitive porous media Zhang andTong [23] investigated the pressure transient response ofthe fractal medium in stress-sensitive reservoirs using theself-similarity solution method and the regular perturbationmethod Marshall [24] presented a new analytical methodfor solving the flow equation which contains a squared gra-dient term and an exponential dependence of the hydraulicdiffusivity on pressure Friedel and Voigt [25] investigatedapproximate analytical solutions for slightly compressiblefluids and real gas flow with constant-rate and constant-pressure wellbore boundary conditions using the perturba-tion technology and Boltzmann transformation Zhang etal [26] presented the well test model for stress-sensitiveand heterogeneous reservoirs with nonuniform thicknessesand obtained the zero-order approximate analytical solutionusing the perturbation technology Zhang et al [27] extendedthe work of Zhang et al [26] to the case of stress-sensitiveand heterogeneous dual-porosity reservoirswith nonuniformthicknesses Qanbari and Clarkson [28] presented a newmethod to solve the nonlinear partial differential equation forthe transient flow of slightly compressible liquid in a stress-sensitive formation by introducing a new variable Ai and Yao[29] presented a new well test model for low-permeability

stress-sensitive reservoirs in which all permeability porosityand pore compressibility are pressure dependent Yi et al[30] presented a mathematical model of fluid flow in astress-sensitive reservoir with a horizontal well based onthe pressure-dependent characteristics of permeability andporosity

Although many models for stress-sensitive reservoirsbased on the permeability-stress function approach havebeen presented almost all published models are only con-fined to the exponential function (or the modified expo-nential function) relationship between the permeability andthe pore pressure (or the effective overburden pressure)Recently through a lot of actual experimental data severalinvestigators [12 31ndash36] have found that the decrease of per-meability as a function of the effective overburden pressurecan be described by a power function which usually fits withexperimental data better than the exponential function Butsurprisingly although the power function model has servedas a good alternative method for describing the relationshipbetween the permeability and the effective overburden pres-sure there have been few attempts to introduce the powerfunction model into the seepage flow model for studying thetransient flow behavior in stress-sensitive reservoirs

In this paper based on the experimental data of actualcore samples we validate that the improved power functionmodel is a good alternative method for describing thepermeability-stress relationship in comparison with the one-parameter exponential function model Then according tothe improved power function model we present a newmathematical model to study the pressure transient behaviorin stress-sensitive reservoirs The mathematical model issolved by the fully implicit finite difference method whichhas been validated by some published analytical solutionsPressure response curves are plotted and the effects ofrelevant parameters on both pressure drawdown and buildupresponses are studied

2 Mechanisms of Permeability Reduction

21 Laboratory Measurements of Stress Effect on Permeabil-ity In order to accurately simulate the stress state of thereservoir rock the permeability test should be conductedin a three-dimension stress condition In this experimentthe hydrostatic core holder is usually used to simulate thechange of the effective overburden pressure By keeping theconfining pressure constant and changing the fluid pressurein the core through the back pressure valve themeasurementcan be conducted at different effective confining pressures tosimulate the effective overburden pressure variation The teststeps are as follows

(1) Determine the initial effective overburden pressureThe initial effective overburden pressure can beexpressed as

119901119894eff = 119901ob minus 119901119894 (1)

Mathematical Problems in Engineering 3

where 119901ob can be accurately obtained through thedensity log or be approximately determined by thefollowing expression

119901ob = [120601120588119908+ (1 minus 120601) 120588

119904] 119892119867 (2)

(2) Put the core into the core holder and then adjust theconfining pressure and the fluid pressure to set theeffective pressure at the initial effective overburdenpressure which can be calculated by (1) The perme-ability at the initial effective overburden pressure canbe measured

(3) Keep the confining pressure constant and decreasethe fluid pressure in the core Wait till a stable stateis reached in the deformation of the core and thenthe permeability can be measured under differenteffective pressures

(4) Repeat Step (3) The relationship between the perme-ability and the effective overburden pressure can beobtained

Five core samples from119883 field and six core samples from119884 field were tested respectively The permeability-effectiveoverburden pressure curves are shown in Figures 1 and 2respectively It is clear that the permeability decreases whenthe effective overburden pressure increases

22 Curve-Fitting Equations for the Permeability-EffectiveOverburden Pressure Relation In order to study fluid flow instress-dependent porous media some kinds of mathematicalequations are used to describe the relationship between thepermeability and the effective overburden pressure

221 One-Parameter Exponential Function Model The one-parameter exponential function model to represent the rela-tionship between the permeability and the effective over-burden pressure is widely used to study the transient flowbehavior in porous media The one-parameter exponentialfunction model is given as [17 18]

119896 = 119896119894sdot exp [minus120574 (119901

119894minus 119901)] = 119896

119894sdot exp [minus120574 (119901eff minus 119901119894eff)] (3)

222 Stepwise Permeability Model Based on the one-parameter exponential function model stepwise perme-ability model to approximately represent the permeabilitymodulus 120574 changing with net confining pressure has beenintroduced The stepwise permeability model based on thestepwise permeability modulus is given as [19 20]

119896 = (119896119894)119895sdot exp [minus120574

119895(119901119895minus 119901)]

(for 119901119895+1

le 119901 le 119901119895 119895 = 0 1 2 119899)

(4)

where

(119896119894)119895= (119896119894)0sdot exp[minus

119895

sum

119897=1

120574119897minus1

(119901119897minus1

minus 119901119897)]

(for 119895 = 1 2 119899)

(5)

with 1199010= 119901119894and (119896

119894)0= 119896119894

05

04

03

02

01

00

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40

Effective overburden pressure (106 Pa)

Core X1Core X2Core X3

Core X4Core X5

Figure 1 Relationship between permeability and effective overbur-den pressure for core samples from119883 field

140

120

100

80

60

40

20

0

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40 45


Core Y1Core Y2Core Y3


Figure 2 Relationship between permeability and effective overbur-den pressure for core samples from 119884 field

Although the stepwise permeability model seems tobe more reasonable than the one-parameter exponentialfunction model for describing the relationship between thepermeability and the effective overburden pressure the step-wise permeability model has not been widely used in practicebecause the critical pressure is difficult to be determinedaccurately

223 Two-Parameter Exponential Function Model Com-pared with the one-parameter exponential function modelthe two-parameter exponential function model to represent


stress-sensitive permeability is proposed to allow for the per-meability to approach a specifiedminimum permeability notnecessarily zero at large effective overburden pressure Thetwo-parameter exponential function model can be expressedas [20 37]

119896 minus 119896infin

119896119894minus 119896infin

= exp [minus120574 (119901119894minus 119901)] = exp [minus120574 (119901eff minus 119901119894eff)] (6)

224 Power Function Model In recent years through a lotof actual experimental data some scholars have found thata power-function curve-fitting equation can mathematicallyrepresent the experimental data for core samples and usuallyobtain a better correlation between the measured data andcalculated values than the exponential-function curve-fittingequation Therefore the power function model which is agood alternativemethod to describe the relationship betweenthe permeability and the effective overburden pressure hasreceivedmore attentionThe power functionmodel is usuallywritten as [12 34]

119896 = 1198960sdot (119901ob minus 119901)

minus119898= 1198960sdot 119901minus119898

eff (7)

225 Improved Power Function Model Although (7) is agood curve-fitting equation used to describe the relationshipbetween the permeability and the effective overburden pres-sure (7) may hinder the study of the fluid flow in stress-dependent porous media because the value of119898 depends onthe unit of 119901eff In order to obtain the dimensionless relation-ship between the permeability and the effective overburdenpressure the improved power function model is proposed asfollows [31ndash33 35]

119896

119896119894

= (119901ob minus 119901

119901ob minus 119901119894)

minus119898

= (119901eff119901119894eff)

minus119898

(8)

Inwhat follows (8) would be used to establish the seepagemodel for studying the fluid flow in stress-dependent porousmedia

23 The Fitting Comparison between These Kinds of Math-ematical Equations The experimental data of actual coresamples from119883field and119884field and the ones given byVairogset al [7] are used to analyze the fitting correlation betweenthe measured data and calculated values from the fittingequations respectively It should be noted that both the powerfunction model and the improved power function model arebased on the power function relation between the perme-ability and the effective overburden pressure Therefore inthe following we focus on the fitting comparison betweenthe one-parameter exponential function model which hasbeen widely applied in studying the transient flow behaviorin porous media and the improved power function model

The experimental data of actual core samples from 119883

field and 119884 field are fitted by (3) and (8) respectively Thefitting parameters (ie stress-sensitive coefficient 119898 and120574) and correlation coefficients 1198772 of each core sample arelisted in Tables 1 and 2 respectively As shown in Tables1 and 2 for core samples from 119883 field and 119884 field most

of correlation coefficients between the measured data andcalculated values from the improved power function modelare greater than the ones from the one-parameter exponentialfunction model and the average correlation coefficientsbetween the measured data and calculated values from theimproved power function model are also greater than theones from the one-parameter exponential function modelrespectively

For further analyzing the fitting comparison betweenone-parameter exponential function model and improvedpower function model the measured data reported byVairogs et al [7] are used to be fitted by (3) and (8)respectively The results including the fitting parameters andcorrelation coefficients are listed in Table 3 which shows thatgood agreement between the measured data and calculatedvalues from the improved power function model is obtainedfor most of core samples and the average correlation coeffi-cient between the measured data and calculated values fromthe improved power function model is greater than the onefrom the one-parameter exponential function model

Consequently the improved power function modelbased on the power function relation between the permeabil-ity and the effective overburden pressure is a good alternativemethod for describing the permeability-stress relationshipin comparison with the one-parameter exponential functionmodel which has been widely used in practice

3 Mathematical Modeling

31 Assumptions An isotropic homogeneous horizontaland slab reservoir is bounded by the top and the bottomparallel impermeable planes The reservoir is filled witha single fluid which is a slightly compressible fluid withconstant viscosity The fluid flow in the reservoir followsDarcyrsquos law with the influence of gravity force and capillaryforce being ignored The initial pressure is assumed to beuniform throughout the reservoir The formation rock isthe sensitivity of permeability to effective stress Fluid isproduced at a constant rate by a finite-radius well withwellbore storage and skin

32 Establishment of Mathematical Model Based on theimproved power functionmodelmentioned above themath-ematical model used to study the transient flow behavior instress-sensitive porous media is derived in Appendix A Forthe convenience of calculation and application the dimen-sionless variables which are defined in Table 4 where all theparameters are explained in the nomenclature are introducedinto the mathematical model Therefore the dimensionlessmathematical model is as follows

Dimensionless seepage flow differential equation is asfollows

1205972119901119863

1205971199032119863

+1

119903119863

120597119901119863

120597119903119863

minus (119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119903119863

)

2

= (119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(9)


Table 1 The fitting comparison between one-parameter exponential function model and improved power function model for core samplesfrom119883 field

Sample name 119896119894 (10minus15m2)

Improved power function model119896119896119894= (119901eff119901119894eff)

minus119898

One-parameter exponential function model119896119896119894= exp[minus120574(119901eff minus 119901119894eff)]

119898 1198772

120574 (10minus6 Paminus1) 1198772

X1 0233 0754 0997 0039 0974X2 0413 0519 0997 0027 0967X3 0030 1079 0996 0057 0977X4 0055 0977 0986 0052 0997X5 0079 0719 0990 0037 0960Average value 0162 0810 0993 0042 0975

Table 2 The fitting comparison between one-parameter exponential function model and improved power function model for core samplesfrom 119884 field



minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772

Y1 019 0385 0943 0020 0888Y2 318 0095 0940 0005 0824Y3 168 0052 0981 0003 0901Y4 468 0038 0977 0002 0885Y5 795 0025 0991 0001 0920Y6 1370 0040 0990 0002 0920Average value 47245 0106 0970 0006 0890

Initial condition is

119901119863

1003816100381610038161003816119905119863=0= 0 (10)

Inner boundary condition for constant-rate production is

119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= 1 (11)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

(12)

Outer boundary conditions are the following

lim119903119863rarrinfin

119901119863= 0 (infinite) (13)

119901119863

1003816100381610038161003816119903119863=119903119890119863= 0 (constant pressure) (14)

(119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (closed) (15)

Based on the mathematical model including (9) to (15)several simplified models can be obtained as follows

(1) When 119898 = 0 and 120572 = 0 the mathematical modelis reduced to the conventional radial flow modelwithout considering the effects of the stress sensitivityand the quadratic gradient term which has beenstudied for a long time [38 39]

(2) When 119898 = 0 and 120572 = 0 the mathematical model isreduced to the radial flow model with only consider-ing the effect of the quadratic gradient term whichhas attracted attention and has been studied since the1990s [40ndash42]

(3) When 119898 = 0 and 120572 = 0 the mathematical model isreduced to the radial flow model with only consid-ering the effect of the stress sensitivity Because ofneglecting the quadratic gradient in the certain oper-ations such as hydraulic fracturing large-drawdownflows drill-stem test and large-pressure pulse testingthis simplified model may cause significant errorof the predicted pore pressure for stress-sensitivereservoirs

It should be noted that the present model may not beavailable for simulating the fluid flows in stress-sensitivereservoirs with lots of very narrow pores (a few nanometerthick pores) That is because the present model is basedon Darcyrsquo law which neglects the microscale effects and isalways employed to simulate macroflows in porous mediawith lots of micrometer pores For the porous media withlots of nanopores it is important to consider the microscaleeffects in simulationsTherefore the presentmodel cannot beapplied to the simulation of fluid flows in porous media withlots of nanopores while the molecular simulation methodcan successfully simulate the fluid flows in nanopore [43ndash45]


Table 3 The fitting comparison between one-parameter exponential function model and improved power function model for core samplesas reported by Vairogs et al [7]



minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772

Core A 191 0033 0952 0001 0995Core B 15 0076 0984 0003 0915Core C 17 0099 0999 0003 0820Core D 0186 0248 0996 0011 0821Core E 004 0494 0962 0040 0986Core F 414 0921 0993 0108 0987Core G 048 0577 0994 0127 0952Core H 015 0940 0995 0121 0980Core I 014 0452 0967 0038 0976Core J 0377 0633 0975 0074 0992Core K 00872 0254 0994 0013 0816Average value 193909 0430 0983 0049 0931

Table 4 Definitions of dimensionless variables

Dimensionless pressure 119901119863=2120587119896119894ℎ (119901119894minus 119901)

119902sc119861120583

Dimensionless overburden pressure 119901ob119863 =2120587119896119894ℎ (119901119894minus 119901ob)

119902sc119861120583

Dimensionless wellbore pressure 119901119908119863

=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583

Dimensionless time 119905119863=

119896119894119905

1206011205831198881199051199032119908

Dimensionless distance 119903119863=

119903

119903119908

119903119890119863=119903119890

119903119908

Dimensionless quadratic gradient coefficient 120572 =119902sc119861120583119888119891

2120587119896119894ℎ

Dimensionless wellbore storage coefficient 119862119863=

119862

2120587120601119888119905ℎ1199032119908

33 Solutions of Mathematical Model Because the seepageflow differential equation in the mathematical model is anonlinear differential equation the analytical solution ishardly obtainedTherefore in this study (9) to (15) are solvednumerically by finite difference method The grid blocks inthe radial direction are spaced in a geometric fashion A fullyimplicitmethod is used to generate the finite-difference formsof (9) to (15) and the Newton-Raphson method is used tosolve the resulting system of nonlinear equationsThe processof solving the model is shown in Appendix B

It should be noted that the time step size and space stepsize are not constants The time step size and space step sizemay be increased or decreased according to the convergenceand accuracy

In practice pressure buildup is widely applied in welltest For pressure buildup response the ldquosuperposition intimerdquo method used for analytical solution of linear differ-ential equation cannot be directly used for stress-sensitivereservoirs because (9) is a nonlinear differential equation

Therefore in order to obtain the pressure buildup responsethe well is directly shut-in after the specified producing timeis reached by setting the right-hand side of (11) and (B16) tobe zero

It is important to validate the numerical solution beforeusing the numerical simulator to compute the pressuretransient response for stress-sensitive reservoirs In the fol-lowing the numerical solution is validated for non-stress-sensitive reservoirs by comparing the dimensionless wellborepressure response obtained by the numerical solution withthe one obtained by the well-known Van Everdingen andHurstrsquos analytical solution [38] The Van Everdingen andHurstrsquos solution was implemented by Ambastha and RameyJr [39] The analytical solution in the Laplace space shouldbe inverted to the real space using the algorithm proposed byStehfest [46]

Figure 3 shows the comparison of numerical pressuredrawdown solution with the Van Everdingen and Hurstanalytical solution for infinite closed and constant pressure


AnalyticalNumerical

Closed outer boundary

Infinite outer boundary

Constant pressure outer boundary

reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD

Figure 3 Comparison of numerical pressure drawdown solutionwith the Van Everdingen and Hurstrsquos analytical solution for 119903

119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD


reD = 1000Infinite outer boundary

S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0

outer boundary conditions considering the effects of wellborestorage and skin The solid lines are analytical results andthe scatter points are numerical results All of them arematched pretty well Figure 4 shows the case in which theeffects of wellbore storage and skin are not considered Thenumerical solutions are also in good agreement with theanalytic solutions

In order to obtain the pressure buildup response theldquosuperposition in timerdquo method is applied in Van Everdingenand Hurstrsquos analytical solution while the ldquodirect shut-inrdquomethod is used in the numerical solution Figure 5 showsthe results of pressure buildup response obtained by the Van

10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD

Figure 5 Comparison of numerical pressure buildup solution withthe Van Everdingen and Hurstrsquos analytical solution for 119903

119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001

Figure 6 Comparison of numerical pressure drawdown solutionwith the Chakrabarty et alrsquos analytical solution for 119903

119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001

Everdingen and Hurstrsquos analytical solution and the numericalsolution A perfect match has been obtained too FromFigures 3ndash5 we can conclude that the numerical solutionin this study agrees with the Van Everdingen and Hurstrsquosanalytical solution very well for both pressure drawdown andbuildup response with or without wellbore storage and skineffects for non-stress-sensitive reservoirs

Chakrabarty et al [40] proposed an analytical solution ofthe radial flow model considering the effect of the quadraticgradient term which is a simplified model of the proposedstress-sensitive model in this study when 119898 = 0 and120572 = 0 The numerical solution is validated by comparing the


10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD

p998400wD middot tDCD

100

101

102

10minus2

10minus1pwD

andp998400 wDmiddottDC

D

m = 120805030

I II

III

Figure 7 Effect of stress-sensitive coefficient 119898 on the wellborepressure transient behavior for an infinite stress-sensitive reservoir

Table 5 Data used in the base case

Initial reservoir pressure 119901119894 Pa 3192 times 10

7

Reservoir depth119867 m 2280

Formation thickness ℎ m 101

Wellbore radius 119903119908 m 011

Fluid viscosity 120583 Pasdots 181 times 10minus3

Fluid compressibility 119888119891 Paminus1 162 times 10

minus9

Pore compressibility 119888119901 Paminus1 75 times 10

minus10

Porosity 120601 dimensionless 01

Volume factor 119861 dimensionless 113

Initial permeability 119896119894 m2

24 times 10minus15

Production rate 119902sc m3s 134 times 10

minus4

Wellbore storage coefficient 119862 m3pa 3 times 10minus8

Skin factor 119878 dimensionless 2

Density of rock skeleton 120588119904 kgm3

2650

Density of formation water 120588119908 kgm3

1000

dimensionless wellbore pressure response obtained by thenumerical solution with the one obtained by the Chakrabartyet alrsquos analytical solution with the effect of the quadraticgradient term The numerical pressure drawdown solutionand the Chakrabarty et alrsquos analytical solution for infiniteclosed and constant pressure outer boundary conditionswith the effect of the quadratic gradient term are plottedin Figure 6 which also shows an excellent match betweenthe numerical solution and the Chakrabarty et alrsquos analyticalsolution

Based on the preceding validation efforts we concludethat the numerical computation method in this study whichhas yielded accurate pressure transient response for non-stress-sensitive reservoirs with or without the effect of thequadratic gradient term can be used to compute the pressuretransient response for stress-sensitive reservoirs

00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD

Figure 8 The relative error between the wellbore pressure for non-stress-sensitive reservoirs and the one for stress-sensitive reservoirswith different stress-sensitive coefficients119898

4 Pressure Transient Characteristics

In this section we will calculate the dimensionless wellborepressure (119901

119908119863) and the derivative (d119901

119908119863d119905119863) for a stress-

sensitive reservoir with the proposed model and numericalcomputation method In what follows the standard log-log typical curves of 119901

119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863

are obtained and the effects of relevant parameters on thepressure transient behavior are studied Basic data used fordemonstration in the base case are shown in Table 5

Figure 7 shows the effect of stress-sensitive coefficient119898on the wellbore pressure transient behavior for an infinitestress-sensitive reservoir As shown in Figure 7 the entiretransient-flow process includes three main flow stages Inearly time (stage I called as wellbore storage period) thepressure curve and the pressure derivative curve which arenot affected by stress sensitivity in this flow period align ina unit slope line Then the transitional flow period (stage II)and the radial flow period (stage III) in which the stress-sensitive coefficient119898 has a significant effect on the pressureand derivative curves can be seen in the typical curves Thepositions of pressure and derivative curves ascend with anincreasing value of 119898 in the transitional flow period and theradial flow period

In order to quantify the effect of stress sensitivity on thepressure behavior we introduce the relative error between thewellbore pressure for non-stress-sensitive reservoirs and theone for stress-sensitive reservoirs with all other parameterskept constant The relative error is expressed as

Relative error

=

1003816100381610038161003816119901119908119863 (stress-sensitive) minus 119901119908119863 (non-stress-sensitive)1003816100381610038161003816

119901119908119863 (non-stress-sensitive)

(16)

Figure 8 shows the relative error between the wellborepressure for non-stress-sensitive reservoirs and the one for


10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500

Figure 9 Effect of the dimensionless outer boundary radius 119903119890119863

on the wellbore pressure transient behavior for stress-sensitivereservoirs with closed outer boundary

10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwDandp998400 wDmiddottDC

D

m = 03

reD = 500 1000 1500

Figure 10 Effect of the dimensionless outer boundary radius119903119890119863 on the wellbore pressure transient behavior for stress-sensitive

reservoirs with constant pressure outer boundary

stress-sensitive reservoirs with different stress-sensitive coef-ficients 119898 As shown in Figure 8 stress sensitivity has noinfluence on the wellbore pressure transient behavior in earlytime (stage I called as wellbore storage period) After thewellbore storage period the relative error increases with timeincreasing The magnitude of the relative error is greatlydependent on the stress-sensitive coefficient 119898 Along withthe increase of 119898 the relative error appears higher In otherwords a larger 119898 value could cause a larger deviation ofwellbore pressure from119898 = 0

Figures 9 and 10 show the effects of the dimensionlessouter boundary radius 119903

119890119863 on thewellbore pressure transient

10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD

Figure 11 Effect of stress-sensitive coefficient 119898 on the wellborepressure buildup for a stress-sensitive reservoir with closed outerboundary

behavior for stress-sensitive reservoirs with closed and con-stant pressure outer boundary respectively It can be seen thatthe outer boundary configurations and the value of 119903

119890119863only

have influence on the wellbore pressure transient behaviorat late time When the reservoir is limited by a closed outerboundary both the pressure and derivative go up at latetime When the reservoir is limited by a constant pressureboundary the pressure stabilizes and the derivative dropstowards zero at late timeThe start time of the outer boundaryreflection is a function of the value of 119903

119890119863 with a large 119903

119890119863 the

outer boundary reflection occurs laterFigure 11 shows the effect of stress-sensitive coefficient

119898 on the wellbore pressure buildup for a stress-sensitivereservoir with closed outer boundary As shown in Figure 11irrespective of the severity of stress sensitivity all wellborepressure buildup responses merge with 119898 = 0 response atlate time It should be noted that the effect of stress sensitivityon the wellbore pressure buildup mainly occurs in early andintermediate time Before the merger of pressure buildupresponses for a stress-sensitive reservoir (ie119898 = 0 response)with the responses for a non-stress-sensitive reservoir (ie119898 = 0 response) in Figure 11 apparent semilog straight linesof slopes higher than that for119898 = 0 responsemay be drawn atintermediate timewhichmay result in underestimated valuesfor initial effective permeability if not considering the effect ofstress sensitivity for a stress-sensitive reservoirThe larger thevalue of 119898 is the more severe the underestimation of initialeffective permeability becomes

5 Conclusions

This paper has presented a new mathematical model forstudying the pressure transient behavior in stress-sensitivereservoirs based on the improved power functionmodelTheproposed model has been solved by the fully implicit finitedifferencemethodThe effects of relevant parameters on bothpressure drawdown and buildup responses have been studied


Themodel presented in this study has provided an alternativemethod for understanding and predicting the performancesfor stress-sensitive reservoirs Several important conclusionscan be drawn from this study

(1) The improved power function model which is basedon the power function relation between the perme-ability and the effective overburden pressure providesa good match to the experimental data of actual coresamples and could serve as a good alternative methodfor describing the permeability-stress relationshipin comparison with the one-parameter exponentialfunction model

(2) The numerical computation method proposed in thisstudy which has been validated by some publishedanalytical solutions can be used to compute thepressure drawdown and buildup responses for stress-sensitive reservoirs

(3) Irrespective of the severity of stress sensitivity thepressure curve and the pressure derivative curvealways align in a unit slope line in early time

(4) After the wellbore storage period stress sensitivityhas an important effect on the wellbore pressuretransient behavior The positions of pressure andderivative curves ascend with an increasing value of119898 The relative error between the wellbore pressurefor non-stress-sensitive reservoirs and the one forstress-sensitive reservoirs increases with time and thevalue of119898 increasing

(5) The outer boundary configurations and the valueof 119903119890119863

only have influence on the wellbore pressuretransient behavior at late time The start time of theouter boundary reflection is a function of the valueof 119903119890119863 with a large 119903

119890119863 the outer boundary reflection

occurs later(6) In all likelihood a conventional semilog analysis of

pressure buildup data from stress-sensitive reservoirsassumed to be falling in the radial flow period willunderestimate the value of initial effective perme-ability The higher the stress sensitivity is the moresevere the underestimation of the initial effectivepermeability becomes

Appendices

A Mathematical Modeling

The continuity equation for a one-dimensional radial systemis given as

1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)

Darcyrsquos law under the above assumptions takes the form

V =119896

120583

120597119901

120597119903 (A2)

The permeability-stress relationship can be expressed by (8)and the fluid compressibility is defined as follows

119888119891=1

120588

120597120588

120597119901 (A3)

The pore compressibility is defined as follows

119888119901=1

120601

120597120601

120597119901 (A4)

Taking (A2) to (A4) and (8) into (A1) the seepage flowdifferential equation for stress-sensitive reservoirs in a one-dimensional radial system is given by the following

1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901

At time 119905 = 0 pressure is distributed uniformly in thereservoir equal to the initial pressure 119901

119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)

The outer boundary may be infinite closed or constantpressure The outer boundary conditions are as follows

lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)

B Solving the Mathematical Model

For the convenience of discretizing the mathematical modellet us introduce a new dimensionless space variable 119909 that isrelated to the dimensionless radial distance according to

119909 = ln 119903119863 (B1)

With the aid of (B1) (9) to (15) can be rewritten as followsrespectively

Dimensionless seepage flow differential equation is

1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863

1003816100381610038161003816119909=119909119890= 0 (constant pressure ) (B7)

(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)

Equations (B2) to (B8) are discretized by a fully implicitmethod and in order to improve the precision of processingthe virtual node will be introduced for the third boundarycondition (eg virtual node minus1 for infinite and constantpressure outer boundary model and virtual node minus1 andvirtual node 119873 + 1 for closed outer boundary model) Thediscrete forms of (B2) to (B8) are as follows

119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)

where 119894 = 0 1 2 119873 minus 1 for infinite outer boundarymodel and constant pressure outer boundary model and 119894 =0 1 2 119873 for closed outer boundary model

1199010

119863119894= 0 (B10)

where 119894 = minus1 0 1 119873 for infinite outer boundary modeland constant pressure outer boundary model and 119894 =

minus1 0 1 119873119873 + 1 for closed outer boundary model

119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (constant pressure)

(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873

Substituting (B12) into (B11) yields

119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)

Equations (B9) (B10) (B16) and different outer boundaries(B13) to (B15) form three discretemodelswith different outerboundaries respectively which are nonlinear equations andcan be solved by the Newton-Raphson method [47]


List of Symbols

Variables

119861 Volume factor dimensionless119862 Wellbore storage coefficient (m3Pa)119862119863 Dimensionless wellbore storage coefficient

dimensionless119888119891 Fluid compressibility (Paminus1)

119888119901 Pore compressibility (Paminus1)

119888119905 Total compressibility (Paminus1)

119892 Gravitational acceleration (ms2)ℎ Reservoir thickness (m)119867 Reservoir depth (m)119896 Permeability (m2)119896119894 Initial permeability (m2)

1198960 Permeability at surface condition (m2)

(119896119894)119895 The 119895th initial permeability corresponding to 120574

119895

(m2)119896infin Limiting value of permeability at infinite stress

(m2)119898 Stress-sensitive coefficient dimensionless119899 Number of total stepwise stress-sensitive

coefficients119873 Number of space grids119901 Reservoir pressure (Pa)119901119863 Dimensionless reservoir pressure

dimensionless119901eff Effective overburden pressure (Pa)119901119894 Initial reservoir pressure (Pa)

119901119894eff Initial effective overburden pressure (Pa)119901119895 The 119895th critical pressure (Pa)

119901ob Overburden pressure (Pa)119901ob119863 Dimensionless overburden pressure

dimensionless119901119908 Wellbore pressure (Pa)

119901119908119863

Dimensionless wellbore pressuredimensionless

119902sc Production rate at surface condition (m3s)119903 Radial distance (m)119903119863 Dimensionless radial distance dimensionless

119903119890 Outer reservoir radius (m)

119903119890119863 Dimensionless outer reservoir radius

dimensionless119903119908 Wellbore radius (m)

119878 Skin factor dimensionless119905 Time (s)119905119863 Dimensionless time dimensionless

V Fluid velocity (ms)119909 Transformed dimensionless variable of 119903

119863

dimensionless119909119890 Transformed dimensionless variable of 119903

119890119863

dimensionless120572 Dimensionless quadratic gradient coefficient

dimensionless120601 Porosity of reservoir fraction120588 Density of fluid (kgm3)

120588119904 Density of rock skeleton (kgm3)

120588119908 Density of formation water (kgm3)

120574 Stress-sensitive coefficient (Paminus1)120574119895 The 119895th stress-sensitive coefficient (Paminus1)

120583 Viscosity (Pasdots)Δ119909 Space step size dimensionlessΔ119905 Time step size dimensionless

Superscript119895 Time level label

Subscripts

119863 Dimensionless119894 Space location labelSc Standard state

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge with gratitude thesupport by National Science and Technology Major Projectof China (Grant no 2008ZX05013)

References

[1] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[2] N H Kilmer N R Morrow and J K Pitman ldquoPressuresensitivity of low permeability sandstonesrdquo Journal of PetroleumScience and Engineering vol 1 no 1 pp 65ndash81 1987

[3] P M T M Schutjens T H Hanssen M H H Hettema etal ldquoCompaction-induced porositypermeability reduction insandstone reservoirs data and model for elasticity-dominateddeformationrdquo SPE Reservoir Evaluation amp Engineering vol 7no 3 pp 202ndash216 2004

[4] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[5] Y Ma S Zhang T Guo G Zhu X Cai and M Li ldquoPetroleumgeology of the Puguang sour gas field in the Sichuan Basin SWChinardquoMarine and PetroleumGeology vol 25 no 4-5 pp 357ndash370 2008

[6] F Cappa Y Guglielmi P Fenart V Merrien-Soukatchoff andA Thoraval ldquoHydromechanical interactions in a fracturedcarbonate reservoir inferred from hydraulic and mechanicalmeasurementsrdquo International Journal of Rock Mechanics andMining Sciences vol 42 no 2 pp 287ndash306 2005

[7] J Vairogs C L Hearn D Dareing and V W RhoadesldquoEffect of rock stress on gas production from low- permeabilityreservoirsrdquo Journal of Petroleum Technology vol 23 pp 1161ndash1167 1971

[8] X J Xiao H Sun Y Han and J Yang ldquoDynamic characteristicevaluation methods of stress sensitive abnormal high pressure


gas reservoirrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition (ATCE rsquo09) pp 2045ndash2057 NewOrleans La USA October 2009

[9] I Fatt and D H Davis ldquoReduction in permeability withoverburden pressurerdquo Transactions of AIME vol 195 p 3291952

[10] R A Farquhar B G D Smart A C Todd D E Tompkins andA J Tweedie ldquoStress sensitivity of low petroleum sandstonesfrom the rotliegendes sandstonerdquo in Proceedings of the 68thSPE Annual Technical Conference and Exhibition pp 851ndash861Houston Tex USA October 1993

[11] Z Zeng R Grigg and D B Gupta ldquoLaboratory investigationof stress- sensitivity of non-Darcy gas flow parametersrdquo in Pro-ceedings of the SPEDOE Symposium on Improved Oil RecoveryTulsa Okla USA April 2004

[12] S L Yang X Q Wang J L Feng and Y X Su ldquoTest and studyof the rock pressure sensitivity for KeLa-2 gas reservoir in theTarim basinrdquo Petroleum Science vol 1 no 4 pp 11ndash16 2004

[13] H H Abass A M Tahini Y N Abousleiman and M KhanldquoNew technique to determine Biot coefficient for stress sensitivedual porosity reservoirsrdquo in SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) pp 2292ndash2301 New Orleans LaUSA October 2009

[14] R Raghavan J D T Scorer and F G Miller ldquoAn investigationby numerical methods of the effect of pressure-dependent rockand fluid properties on well flow testsrdquo SPE Journal vol 12 no3 pp 267ndash275 1972

[15] V F Samaniego W E Brigham and F G Miller ldquoAn investiga-tion of transient flow of reservoir fluids considering pressure-dependent rock and fluid propertiesrdquo SPE Journal vol 17 no 2pp 140ndash150 1977

[16] V F Samaniego and H Cinco-Ley ldquoOn the determination ofthe pressure-dependent characteristics of a reservoir throughtransient pressure testingrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition pp 20ndash19774 San Anto-nio Tex USA October 1989

[17] O A Pedrosa ldquoPressure transient response in stress-sensitiveformationsrdquo in Proceedings of the SPE California RegionalMeeting Oakland Calif USA

[18] J Kikani and O A Pedrosa Jr ldquoPerturbation analysis of stress-sensitive reservoirsrdquo SPE Formation Evaluation vol 6 no 3 pp379ndash386 1991

[19] M Y Zhang and A K Ambastha ldquoNew insights in pressure-transient analysis for stress-sensitive reservoirsrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition pp 617ndash628 New Orleans La USA September 1994

[20] A K Ambastha and M Y Zhang ldquoIterative and numeri-cal solutions for pressure-transient analysis of stress-sensitivereservoirs and aquifersrdquo Computers amp Geosciences vol 22 no6 pp 601ndash606 1996

[21] T A Jelmert and H Selseng ldquoPressure transient behavior ofstress-sensitive reservoirsrdquo in Proceedings of the Latin Ameri-can and Caribbean Petroleum Engineering Conference Rio deJaneiro Brazil 1997

[22] Y-S Wu and K Pruess ldquoIntegral solutions for transient fluidflow through a porous medium with pressure-dependent per-meabilityrdquo International Journal of Rock Mechanics and MiningSciences vol 37 no 1-2 pp 51ndash61 2000

[23] Y G Zhang and D K Tong ldquoThe pressure transient analysis ofdeformation of fractal mediumrdquo Journal of Hydrodynamics Bvol 20 no 3 pp 306ndash313 2008

[24] 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

[25] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference pp 95ndash113 Denver Colo USA 2009

[26] L Zhang J Guo and Q Liu ldquoA well test model for stress-sensitive andheterogeneous reservoirswith non-uniform thick-nessesrdquo Petroleum Science vol 7 no 4 pp 524ndash529 2010

[27] L-H Zhang J-J Guo andQ-G Liu ldquoA newwell test model forstress-sensitive and radially heterogeneous dual-porosity reser-voirs with non-uniform thicknessesrdquo Journal of HydrodynamicsB vol 23 no 6 pp 759ndash766 2011

[28] F Qanbari and C R Clarkson ldquoRate transient analysis ofstress-sensitive formations during transient linear flow periodrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Canada 2012

[29] S Ai and Y D Yao ldquoFlow model for well test analysis of low-permeability and stress-sensitive reservoirsrdquo Special Topics andReviews in Porous Media vol 3 no 2 pp 125ndash138 2012

[30] X Y Yi Z Zhang C Y Li D C Li and S B Wang ldquoWell testinterpretation method of horizontal well in stress-sensitive gasreservoirrdquo Applied Mechanics and Materials vol 295 pp 3183ndash3191 2013

[31] R L Luo A study of deformation and percolation mechanismsof deep gas reservoir and its application [PhD thesis] ChinaUniversity of Petroleum Beijing China 2006

[32] R-L Luo L-S Cheng J-C Peng andH-Y Zhu ldquoNewmethodof determining relationship between permeability and effectiveoverburden pressure for low-permeability reservoirsrdquo Journal ofChina University of Petroleum vol 31 no 2 pp 87ndash90 2007

[33] R L Luo and J D Feng ldquoDeformation characteristics of lowpermeability rocks under confining pressurerdquo in Proceedingsof the International Forum on Porous Flow and ApplicationsWuhan China 2009

[34] N Zisser and G Nover ldquoAnisotropy of permeability andcomplex resistivity of tight sandstones subjected to hydrostaticpressurerdquo Journal of Applied Geophysics vol 68 no 3 pp 356ndash370 2009

[35] X W Wang Study on micro pore structure and seepage flow inultra-low permeability of Daqing oilfield [PhD thesis] ChineseAcademy of Sciences Beijing China 2010

[36] G Xiao M Jingjing C Yiwei and L Weixin ldquoAn integrationapproach for evaluating well deliverability in ultra deep sandsrdquoin Proceedings of the SPE Production and Operations Conferenceand Exhibition pp 177ndash198 Tunis Tunisia June 2010

[37] J F Evers and E Soeiinah ldquoTransient tests and long-range per-formance predictions in stress-sensitive gas reservoirsrdquo Journalof Petroleum Technology vol 29 no 8 pp 1025ndash1030 1977

[38] A F van Everdingen and W Hurst ldquoThe application of theLaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[39] A K Ambastha andH J Ramey Jr ldquoWell-test analysis for awellin a finite circular reservoirrdquo Journal of Canadian PetroleumTechnology vol 32 no 6 pp 34ndash38 1993

[40] C Chakrabarty S M F Ali and W S Tortike ldquoAnalyticalsolutions for radial pressure distribution including the effectsof the quadratic-gradient termrdquo Water Resources Research vol29 no 4 pp 1171ndash1177 1993


[41] S Braeuning T A Jelmert and S A Vik ldquoThe effect of thequadratic gradient term on variable-rate well-testsrdquo Journal ofPetroleum Science and Engineering vol 21 no 3-4 pp 203ndash2221998

[42] X L Cao D K Tong and R H Wang ldquoExact solutions fornonlinear transient flow model including a quadratic gradienttermrdquo Applied Mathematics and Mechanics vol 25 no 1 pp102ndash109 2004

[43] H Eslami F Mozaffari J Moghadasi and F Muller-PlatheldquoMolecular dynamics simulation of confined fluids inisosurface-isothermal- isobaric ensemblerdquo The Journal ofChemical Physics vol 129 no 19 Article ID 194702 2008

[44] H Eslami and F Muller-Plathe ldquoMolecular dynamics simu-lation of water influence on local structure of nanoconfinedpolyamide-66rdquoThe Journal of Physical Chemistry B vol 115 no32 pp 9720ndash9731 2011

[45] N Mehdipour N Mousavian and H Eslami ldquoMoleculardynamics simulation of the diffusion of nanoconfined fluidsrdquoJournal of the Iranian Chemical Society vol 11 no 1 pp 47ndash522014

[46] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[47] J H Mathews and K D Fink Numerical Methods UsingMATLAB Prentice Hall Upper Saddle River NJ USA 1999

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The permeability-stress function approach consideringthe pressure-dependent permeability as a permeability-stressfunction (eg exponential function) is widely used to studythe transient flow behavior in porous media through solvinga nonlinear diffusivity equation Based on the exponentialfunction relationship between the permeability and the porepressure (or the effective overburden pressure) Pedrosa [17]presented the first-order approximate analytical solution fora line-source well producing at a constant rate from aninfinite radial flow system using the perturbation techniqueKikani and Pedrosa [18] extended the work of Pedrosa [17] bypresenting the second-order approximate analytical solutionfor the same problem using the same solution method-ology and suggested the use of zero-order perturbationsolution to investigate the effects of wellbore storage skinand outer boundary on the pressure transient response forstress-sensitive reservoirs Zhang and Ambastha [19] furtherinvestigated the effects of stress-sensitive permeability ondrawdown and buildup pressure transient behavior using thestepwise permeability model and found that the pressuretransient response obtained by Kikani and Pedrosa [18] wasnot accurate when wellbore storage and skin were takeninto consideration so the numerical method was suggestedto study the pressure transient response in stress-sensitivereservoirs Ambastha and Zhang [20] extended the workof Zhang and Ambastha [19] by presenting three models(ie one-parametermodel stepwise permeabilitymodel andtwo-parameter model) for stress-sensitive reservoirs basedon the exponential function (or the modified exponentialfunction) relationship between the permeability and the porepressure (or the effective overburden pressure) Jelmert andSelseng [21] presented an exponential permeability modelthrough introducing normalized permeability variables andobtained approximate analytical solutions which are compu-tationally simple and readily available Wu and Pruess [22]presented an integral method for describing the transientflow behavior in stress-sensitive porous media Zhang andTong [23] investigated the pressure transient response ofthe fractal medium in stress-sensitive reservoirs using theself-similarity solution method and the regular perturbationmethod Marshall [24] presented a new analytical methodfor solving the flow equation which contains a squared gra-dient term and an exponential dependence of the hydraulicdiffusivity on pressure Friedel and Voigt [25] investigatedapproximate analytical solutions for slightly compressiblefluids and real gas flow with constant-rate and constant-pressure wellbore boundary conditions using the perturba-tion technology and Boltzmann transformation Zhang etal [26] presented the well test model for stress-sensitiveand heterogeneous reservoirs with nonuniform thicknessesand obtained the zero-order approximate analytical solutionusing the perturbation technology Zhang et al [27] extendedthe work of Zhang et al [26] to the case of stress-sensitiveand heterogeneous dual-porosity reservoirswith nonuniformthicknesses Qanbari and Clarkson [28] presented a newmethod to solve the nonlinear partial differential equation forthe transient flow of slightly compressible liquid in a stress-sensitive formation by introducing a new variable Ai and Yao[29] presented a new well test model for low-permeability

stress-sensitive reservoirs in which all permeability porosityand pore compressibility are pressure dependent Yi et al[30] presented a mathematical model of fluid flow in astress-sensitive reservoir with a horizontal well based onthe pressure-dependent characteristics of permeability andporosity

Although many models for stress-sensitive reservoirsbased on the permeability-stress function approach havebeen presented almost all published models are only con-fined to the exponential function (or the modified expo-nential function) relationship between the permeability andthe pore pressure (or the effective overburden pressure)Recently through a lot of actual experimental data severalinvestigators [12 31ndash36] have found that the decrease of per-meability as a function of the effective overburden pressurecan be described by a power function which usually fits withexperimental data better than the exponential function Butsurprisingly although the power function model has servedas a good alternative method for describing the relationshipbetween the permeability and the effective overburden pres-sure there have been few attempts to introduce the powerfunction model into the seepage flow model for studying thetransient flow behavior in stress-sensitive reservoirs

In this paper based on the experimental data of actualcore samples we validate that the improved power functionmodel is a good alternative method for describing thepermeability-stress relationship in comparison with the one-parameter exponential function model Then according tothe improved power function model we present a newmathematical model to study the pressure transient behaviorin stress-sensitive reservoirs The mathematical model issolved by the fully implicit finite difference method whichhas been validated by some published analytical solutionsPressure response curves are plotted and the effects ofrelevant parameters on both pressure drawdown and buildupresponses are studied

2 Mechanisms of Permeability Reduction

21 Laboratory Measurements of Stress Effect on Permeabil-ity In order to accurately simulate the stress state of thereservoir rock the permeability test should be conductedin a three-dimension stress condition In this experimentthe hydrostatic core holder is usually used to simulate thechange of the effective overburden pressure By keeping theconfining pressure constant and changing the fluid pressurein the core through the back pressure valve themeasurementcan be conducted at different effective confining pressures tosimulate the effective overburden pressure variation The teststeps are as follows

(1) Determine the initial effective overburden pressureThe initial effective overburden pressure can beexpressed as

119901119894eff = 119901ob minus 119901119894 (1)



119901ob = [120601120588119908+ (1 minus 120601) 120588

119904] 119892119867 (2)







119896 = 119896119894sdot exp [minus120574 (119901

119894minus 119901)] = 119896



119896 = (119896119894)119895sdot exp [minus120574

119895(119901119895minus 119901)]

(for 119901119895+1

le 119901 le 119901119895 119895 = 0 1 2 119899)

(4)

where

(119896119894)119895= (119896119894)0sdot exp[minus

119895

sum

119897=1

120574119897minus1

(119901119897minus1

minus 119901119897)]

(for 119895 = 1 2 119899)

(5)

with 1199010= 119901119894and (119896

119894)0= 119896119894

05

04

03

02

01

00

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40



Core X4Core X5


140

120

100

80

60

40

20

0

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40 45










119896119894minus 119896infin



119896 = 1198960sdot (119901ob minus 119901)

minus119898= 1198960sdot 119901minus119898

eff (7)


119896

119896119894

= (119901ob minus 119901

119901ob minus 119901119894)

minus119898

= (119901eff119901119894eff)

minus119898

(8)












1205972119901119863

1205971199032119863

+1

119903119863

120597119901119863

120597119903119863

minus (119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119903119863

)

2

= (119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(9)





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772



119901119863

1003816100381610038161003816119905119863=0= 0 (10)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= 1 (11)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

(12)



119901119863= 0 (infinite) (13)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (closed) (15)










minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772




119902sc119861120583


119902sc119861120583


=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583


119896119894119905

1206011205831198881199051199032119908


119903

119903119908

119903119890119863=119903119890

119903119908


2120587119896119894ℎ


119862

2120587120601119888119905ℎ1199032119908








AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901ob = [120601120588119908+ (1 minus 120601) 120588

119904] 119892119867 (2)







119896 = 119896119894sdot exp [minus120574 (119901

119894minus 119901)] = 119896



119896 = (119896119894)119895sdot exp [minus120574

119895(119901119895minus 119901)]

(for 119901119895+1

le 119901 le 119901119895 119895 = 0 1 2 119899)

(4)

where

(119896119894)119895= (119896119894)0sdot exp[minus

119895

sum

119897=1

120574119897minus1

(119901119897minus1

minus 119901119897)]

(for 119895 = 1 2 119899)

(5)

with 1199010= 119901119894and (119896

119894)0= 119896119894

05

04

03

02

01

00

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40



Core X4Core X5


140

120

100

80

60

40

20

0

Perm

eabi

lity

(10minus15

m2)

0 5 10 15 20 25 30 35 40 45










119896119894minus 119896infin



119896 = 1198960sdot (119901ob minus 119901)

minus119898= 1198960sdot 119901minus119898

eff (7)


119896

119896119894

= (119901ob minus 119901

119901ob minus 119901119894)

minus119898

= (119901eff119901119894eff)

minus119898

(8)












1205972119901119863

1205971199032119863

+1

119903119863

120597119901119863

120597119903119863

minus (119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119903119863

)

2

= (119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(9)





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772



119901119863

1003816100381610038161003816119905119863=0= 0 (10)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= 1 (11)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

(12)



119901119863= 0 (infinite) (13)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (closed) (15)










minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772




119902sc119861120583


119902sc119861120583


=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583


119896119894119905

1206011205831198881199051199032119908


119903

119903119908

119903119890119863=119903119890

119903119908


2120587119896119894ℎ


119862

2120587120601119888119905ℎ1199032119908








AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896119894minus 119896infin



119896 = 1198960sdot (119901ob minus 119901)

minus119898= 1198960sdot 119901minus119898

eff (7)


119896

119896119894

= (119901ob minus 119901

119901ob minus 119901119894)

minus119898

= (119901eff119901119894eff)

minus119898

(8)












1205972119901119863

1205971199032119863

+1

119903119863

120597119901119863

120597119903119863

minus (119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119903119863

)

2

= (119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(9)





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772



119901119863

1003816100381610038161003816119905119863=0= 0 (10)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= 1 (11)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

(12)



119901119863= 0 (infinite) (13)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (closed) (15)










minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772




119902sc119861120583


119902sc119861120583


=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583


119896119894119905

1206011205831198881199051199032119908


119903

119903119908

119903119890119863=119903119890

119903119908


2120587119896119894ℎ


119862

2120587120601119888119905ℎ1199032119908








AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772





minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772



119901119863

1003816100381610038161003816119905119863=0= 0 (10)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= 1 (11)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

(12)



119901119863= 0 (infinite) (13)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (closed) (15)










minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772




119902sc119861120583


119902sc119861120583


=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583


119896119894119905

1206011205831198881199051199032119908


119903

119903119908

119903119890119863=119903119890

119903119908


2120587119896119894ℎ


119862

2120587120601119888119905ℎ1199032119908








AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus119898


119898 1198772

120574 (10minus6 Paminus1) 1198772




119902sc119861120583


119902sc119861120583


=2120587119896119894ℎ (119901119894minus 119901119908)

119902sc119861120583


119896119894119905

1206011205831198881199051199032119908


119903

119903119908

119903119890119863=119903119890

119903119908


2120587119896119894ℎ


119862

2120587120601119888119905ℎ1199032119908








AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










AnalyticalNumerical




reD = 1000

CD = 10

S = 2

20

15

10

5

0

pwD

10minus1 100 101 102 103 104 105 106 107 108 109

tD


119890119863=

1000 119862119863= 10 and 119878 = 2

20

15

10

5

0

pwD



S = 0


101 102 103 104 105 106 107 108 109

tD

AnalyticalNumerical

CD = 0


119890119863=

1000 119862119863= 0 and 119878 = 0



10minus1 100 101 102 103 104 105 1060

2

4

6

8

10

AnalyticalNumerical

reD = 100

CD = 10

S = 2

pwD

tpD = 103

tpD = 104

ΔtD


119890119863= 100

119862119863= 10 and 119878 = 2

0

5

10

15

20




AnalyticalNumerical

10minus1 100 101 102 103 104 105 106 107 108 109

tD

pwD

reD = 1000

CD = 10

S = 0

a = 001


119890119863= 1000 119862

119863=

10 119878 = 0 and 120572 = 001




10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 120805030

I II

III




7






minus9


minus10




24 times 10minus15


minus4




2650


1000



00

01

02

03

04

05

06

Rela

tive e

rror m = 12

100 101 102 103 104 105 106 107 108

m = 03

m = 05

m = 08

tD





119908119863d119905119863) for a stress-


119908119863and (119901

1015840

119908119863sdot 119905119863119862119863) versus 119905

119863119862119863




Relative error

=



(16)



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2

10minus1pwD


D

m = 03

reD = 500 1000 1500



10minus2 10minus1 100 101 102 103 104 105 106 107 108

pwD

tDCD


100

101

102

10minus2


D

m = 03

reD = 500 1000 1500






10minus1 100

2

4

6

8

10

0 101 102 103 104 105 106

pwD

m = 10

m = 05

m = 0

reD = 1000

tpD = 104

ΔtD



119890119863only


119890119863 with a large 119903

119890119863 the



5 Conclusions













Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Appendices



1

119903

120597 (119903120588V)120597119903

=120597 (120588120601)

120597119905 (A1)


V =119896

120583

120597119901

120597119903 (A2)


119888119891=1

120588

120597120588

120597119901 (A3)


119888119901=1

120601

120597120601

120597119901 (A4)


1205972119901

1205971199032+1

119903

120597119901

120597119903+ (

119898

119901ob minus 119901+ 119888119891)(

120597119901

120597119903)

2

=119888119905120583120601

119896119894

(119901ob minus 119901

119901ob minus 119901119894)

119898120597119901

120597119905

(A5)

where 119888119905= 119888119891+ 119888119901


119894


1199011003816100381610038161003816119905=0 = 119901

119894 (A6)


119862d119901119908

d119905minus2120587119896119894ℎ

120583(119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

= minus119902sc119861

119901119908= [119901 minus 119878(

119901ob minus 119901

119901ob minus 119901119894)

minus119898

119903120597119901

120597119903]

100381610038161003816100381610038161003816100381610038161003816119903=119903119908

(A7)


lim119903rarrinfin

119901 = 119901119894 (infinite)

1199011003816100381610038161003816119903=119903119890

= 119901119894

(constant pressure)

(119901ob minus 119901

119901ob minus 119901119894)

minus119898120597119901

120597119903

100381610038161003816100381610038161003816100381610038161003816119903=119903119890

= 0 (closed)

(A8)



119909 = ln 119903119863 (B1)



1205972119901119863

1205971199092minus (

119898

119901119863minus 119901ob119863

+ 120572)(120597119901119863

120597119909)

2

= 1198902119909(119901ob119863 minus 119901119863119901ob119863

)

119898120597119901119863

120597119905119863

(B2)



119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901119863

1003816100381610038161003816119905119863=0= 0 (B3)


119862119863

d119901119908119863

d119905119863

minus (119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0

= 1 (B4)

119901119908119863

= [119901119863minus 119878(

119901ob119863 minus 119901119863119901ob119863

)

minus119898120597119901119863

120597119909]

100381610038161003816100381610038161003816100381610038161003816119909=0

(B5)


lim119909rarrinfin

119901119863= 0 (infinite) (B6)

119901119863


(119901ob119863 minus 119901119863119901ob119863

)

minus119898

119890minus119909 120597119901119863

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=119909119890

= 0 (closed) (B8)


119901119895+1

119863119894minus1minus 2119901119895+1

119863119894+ 119901119895+1

119863119894+1

(Δ119909)2

= (119898

119901119895+1

119863119894minus 119901ob119863

+ 120572)(119901119895+1

119863119894minus 119901119895+1

119863119894minus1

Δ119909)

2

+ 1198902119894Δ119909

(119901ob119863 minus 119901

119895+1

119863119894

119901ob119863)

119898

119901119895+1

119863119894minus 119901119895

119863119894

Δ119905

(119895 = 0 1 2 )

(B9)


1199010

119863119894= 0 (B10)



119862119863

119901119895+1

119908119863minus 119901119895

119908119863

Δ119905minus (

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909= 1

(119895 = 0 1 2 )

(B11)

119901119895+1

wD = 119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

(119895 = 0 1 2 )

(B12)

119901119895+1

119863119894= 0 (119894 = 119873 119895 = 0 1 2 ) (infinite) (B13)

119901119895+1


(B14)

(119901ob119863 minus 119901

119895+1

119863119873

119901ob119863)

minus119898

119890minus119873Δ119909

119901119895+1

119863119873+1minus 119901119895+1

119863119873minus1

2Δ119909= 0

(119895 = 0 1 2 ) (closed)

(B15)

where Δ119909 = 119909119890119873 = ln(119903

119890119863)119873


119862119863

Δ119905[

[

119901119895+1

1198630minus 119878(

119901ob119863 minus 119901119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909]

]

minus (119901ob119863 minus 119901

119895+1

1198630

119901ob119863)

minus119898

119901119895+1

1198631minus 119901119895+1

119863minus1

2Δ119909

minus119862119863

Δ119905[

[

119901119895

1198630minus 119878(

119901ob119863 minus 119901119895

1198630

119901ob119863)

minus119898

119901119895

1198631minus 119901119895

119863minus1

2Δ119909]

]

= 1

(119895 = 0 1 2 )

(B16)



List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










List of Symbols

Variables








119895








119901119908119863








119863


119890119863








Subscripts




Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new mathematical model for pressure...

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