Research ArticleA Mathematical Model for the Analysis of the Pressure TransientResponse of Fluid Flow in Fractal Reservoir

Jin-Zhou Zhao1 Cui-Cui Sheng1 Yong-Ming Li1 and Shun-Chu Li2

1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University Chengdu 610500 China2Institute of Applied Mathematics Xihua University Chengdu 610039 China

Correspondence should be addressed to Shun-Chu Li lishunchu163com

Received 12 August 2014 Accepted 14 October 2014

Academic Editor Jianchao Cai

Copyright copy 2015 Jin-Zhou Zhao et alThis 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

This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractalreservoir which can avoid the complicated mathematical deduction The models are presented in three kinds of outer boundaryconditions (infinite constant pressure and closed) The influence of wellbore storage effect skin factor and variable flow rateproduction is also involved in the inner boundary conditions The analytical solutions are constructed in the Laplace space andpresented in a pattern with one continued fractionmdashthe similar structure of solutionThe pattern can bring convenience to well testanalysis programmingThemathematical beauty of fractal is that the infinite complexity is formed with relatively simple equationsSo the relation of reservoir parameters (wellbore storage effect the skin factor fractal dimension and conductivity index) theformation pressure and the wellbore pressure can be learnt easily Type curves of the wellbore pressure and pressure derivative areplotted and analyzed in real domain using the Stehfest numerical invention algorithm The SCMS and type curves can interpretintuitively transient pressure response of fractal spherical flow reservoir The results obtained in this study have both theoreticaland practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS

1 Introduction

The mechanics of oil and gas seepage is a discipline whichresearches the law and state of fluid flow in porous mediaThe practical development of oil and gas reservoirs showsthat reservoir distribution and its space structure are awfullycomplicated In 1982 Mandelbrot and Blumen first pro-posed the fractal geometry theory which used self-similar tocharacterize the complexity of things [1] By combining thetheories of fractal and seepagemechanics it is able to describefluid flow paths in porous media effectively Acuna et alexplained the fractal characters of natural porous media withextremely complex pore structure [2] Because flow paths offluid in porous media can be seen as tortuous capillaries Caiand Yu applied a fractal dimension to describe the tangleof capillary pathways and derived calculation formula ofactual length of tortuous flow paths [3] And spontaneousimbibition of wetting liquid in fractal porousmedia including

gravity was studied [4] The researches were regarded asa crucially important driving mechanism for enhancing oilrecovery in naturally fractured reservoir In the early 1990sChang and Yortsos applied the fractal theory in reservoirmodels and then set up new mathematical models [5 6]Tian and Tong established radial fluid flow models of fractalreservoirs [7] The models have shown that the order of thefractional dimension has influence on the whole pressurebehavior Particularly the effect on pressure behavior is largerin the early time stage Based on fractal geometry and thesemiempirical Kozeny-Carman equation which is the mostfamous permeability-porosity relation in the field of flowin porous media Xu and Yu [8] developed a new formof permeability and Kozeny-Carman constant After thatthey showed that fractal dimension had significant effect onpermeability which can enhance the effective permeability[9 10] Santizo discussed the effect of fractal dimensionand fractal conductivity index on pressure derivative in

Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 596597 8 pageshttpdxdoiorg1011552015596597

2 Journal of Chemistry

finite-conductivity fractures [11] The papers obtained somenumerical or analytical solutions of the formation pressureand the wellbore pressure in different fractal reservoirs orplotted pressure-time curves to analyze the influence ofreservoir parameters [12ndash17]

Different versions of cylindrical flow model have beenstudied on the assumption that wells were completely openedand fully penetrated the productive formation However theassumptionmay seem too restrictive for practical applicationReservoirs have vertical permeability and the length ofinjection or extraction region was small compared to thickformations spherical flowmodel can provide a good approx-imation in practice [18] Schroth and Istok [19] illustratedthe applicability of the derived spherical flow solution andprovided a comparison with its cylindrical flow counterpartThey showed the spherical flow solution increased withincreasing anisotropy in hydraulic conductivities Joseph andKoederitz [20] presented short-time interpretation methodsfor radial-spherical flow in homogeneous and isotropic reser-voirs inclusive of wellbore storage wellbore phase redistribu-tion and damage skin effects Liu [21] studied the transientspherical flow behavior in porous media and derived its non-linear partial differential equation and obtained its analyticalasymptotic and approximate solutions by using the methodsof Laplace transform

In the above proposed studies the solution processes oftheir mathematical models are very complicated Based onsome study in a boundary value problem of the second-order linear homogeneous differential equation Li andLiao recently introduced a new idea similar constructionmethod The similar structure theory and its applicationwere developed maturely [22] Chen and Li presented similarconstruction method of solution (SCMS) for the boundaryvalue problem of Bessel equation and gave the method toanalyze the structure characteristics of solution [23] Afterthat Sheng et al proposed SCMS for a fractal reservoir andlisted its steps [17] They verified that the SCMS is a straight-forward method to solve mathematical models in reservoirengineering but they just did a theoretical study SCMS of theboundary value problems of Airy equation Weber equationEuler hypergeometric equation and composite first Webersystem were studied [24ndash28] SCMS in fractal dual-porosityreservoir and dual-permeability reservoir were presented[29ndash31]

This paper presents amathematicalmodel for the analysisof the pressure transient response of fluid spherical flowin fractal reservoir which considers wellbore storage andskin effect in inner boundary conditions According to amathematical method called SCMS the expressions of thedimensionless formation pressure and the dimensionlesswellbore pressure are constructed in the Laplace space whichavoid complicated calculations Type curves of the wellborepressure and pressure derivative responses are plotted andanalyzed in real domain using the Stehfest numerical inven-tion algorithm [32] The results obtained in this study haveessential significance to understand the pressure character-istics and provide theoretical basis in such a reservoir Thispaper embodies the relationship between engineering andmathematics

Well

Reservoir

Figure 1 Schematic for spherical flow in a reservoir

2 Materials and Methods

21 Reservoir Characteristics Nonpenetrating wells thatoccur in a thick formation can be treated as spherical systemsas shown in Figure 1 Using fractal theory to describe the porenature and transport properties we assume that dimensionalfractal network 119889119891 is embedded in the dimensional Euclideanrock 119889 The fluid only flows in the fractal network andthe flow will obey Darcyrsquos law from the reservoir into thewellbore The reservoir has a uniform thickness of ℎ andoriginal formation pressure is 119901119894 The fractal reservoir ismined with a single small opening hole in the top part Thefluid flow rate is 119902(119905) The gravity and the capillary pressureare ignored

22 Dimensionless Mathematical Model The dimensionlessmathematical model of the reservoir is made up of four partssuch as continuity equation initial condition inner bound-ary condition and outer boundary condition Appendix Apresents mathematical model of fluid flow and dimensionlessvariables

The continuity equation of fluid spherical flow in a fractalreservoir is

1205972119901119863

1205971199032119863

+120573

119903119863

120597119901119863

120597119903119863

= 119903120579119863

120597119901119863

120597119905119863

(1)

Initial condition

119901119863 (119903119863 0) = 0 (2)

Journal of Chemistry 3

Inner boundary condition

(1199032119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

= minus 119902119863 (119905119863) + 119862119863

119889119901119908119863

119889119905119863

119901119908119863 (119905119863) = (119901119863 minus 119878119903119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

(3)

Three kinds of outer boundary conditions are the follow-ing cases

Case 1 Infinite outer boundary condition

119901119863 (infin 119905119863) = 0 (4)

Case 2 Constant pressure outer boundary condition

119901119863 (119877119863 119905119863) = 0 (5)

Case 3 Closed outer boundary condition120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0 (6)

23 A Boundary Value Problem of the Modified Bessel Equa-tion The dimensionless mathematical model ((1)ndash(6)) hap-pens to be a boundary value problem of a second-orderpartial differential equation It can be transformed into aboundary value problem of a second-order ordinary differ-ential equation based on Laplace transform and then it canbe further simplified as a boundary value problem of themodified Bessel equation

1199032119863

1198892119901119863

1198891199032119863

+ 120573119903119863

119889119901119863

119889119903119863

minus 119903120579+2119863 119911119901119863 = 0 (7)

[minus119862119863119911119901119908119863 (119903119863 119911) + (1 + 119862119863119911119878)119889119901119863 (119903119863 119911)

119889119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= minus119902119863 (119911)

(8)

119901119863 (infin 119911) = 0 or 119901119863 (119877119863 119911) = 0 or119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(9)

24 Solution The SCMS is based on a boundary valueproblem of a second-order linear differential equation and isfirstly proposed by Sheng et al [17] Using coefficients of itsright and left boundary condition and its two linearly inde-pendent solutions of the differential equation the solutionof the boundary value problem can be obtained Accordingto the steps of SCMS the solution of the dimensionlessmathematical model of fractal spherical flow reservoir can beconstructed The details are as follows

Step 1 Solve the linearly independent solutions of (7)

119903(1minus120573)2119863 119868] (

2radic119911

2 + 120579119903(2+120579)2119863 )

119903(1minus120573)2119863 119870] (

2radic119911

2 + 120579119903(2+120579)2119863 )

(10)

where 119868](sdot) and 119896V(sdot) denote the first and the secondmodifiedBessel function of order ] = (1 minus 120573)(2 + 120579) respectively

Step 2 Construct a binary function 12059300(1199031198631 1199031198632) employingthe linearly independent solutions (10) and calculate itspartial derivatives for 1199031198631 1199031198632 respectively

12059300 (1199031198631 1199031198632) = (11990311986311199031198632)(1minus120573)2

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(11)

where

120595119898119899 (119909120576 119910120576 120585) = 119870119898 (120585119909

120576) 119868119899 (120585119910

120576)

+ (minus1)119898minus119899+1

119868119898 (120585119909120576)119870119899 (120585119910

120576)

(12)

with real numbers119898 119899

12059301 (1199031198631 1199031198632) =120597

1205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903(1minus120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ radic119911119903(1minus120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059310 (1199031198631 1199031198632) =120597

1205971199031198631

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903minus(1minus120573)21198631 119903

(1minus120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus radic119911119903(1minus120573+120579)21198631 119903

(1minus120573)21198632

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059311 (1199031198631 1199031198632) =1205972

12059711990311986311205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573)2119903minus(1+120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ (1 minus 120573)radic119911119903minus(1+120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus (1 minus 120573)radic119911119903(1minus120573+120579)21198631 119903

minus(1+120573)21198632

4 Journal of Chemistry

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus 119911119903(1minus120573+120579)21198631 119903

(1minus120573+120579)21198632

times 120595]+1]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(13)

Step 3 Construct similar kernel functionsΦ(119903119863 119911) using theouter boundary conditions (9) the binary function equation(11) and its partial derivatives equations (13)

Φ(119903119863 119911) =

12059300 (119903119863infin)

12059310 (1infin) 119901119863 (infin 119911) = 0

12059300 (119903119863 119877119863)

12059310 (1 119877119863) 119901119863 (119877119863 119911) = 0

12059301 (119903119863 119877119863)

12059311 (1 119877119863)

119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(14)

Step 4 Construct the similar structure of solution withthe similar kernel function equation (14) and coefficients(119902119863(119911) 119862119863119911 119878) of the inner boundary condition equation (8)

119901119863 (119903119863 119911) = 119902119863 (119911)1

119862119863119911 + (1 (119878 + Φ (1 119911)))

sdot1

119878 + Φ (1 119911)sdot Φ (119903119863 119911)

(15)

Equation (15) is the analytical solution of the mathemati-cal model of fluid spherical flow in a fractal reservoir Substi-tute (15) into the second expression of (3) the dimensionlesswellbore pressure in the Laplace space can be obtained

119901119908119863 (119911) = 119902119863 (119911)1

119862119863119911 + Φ (1 119911) (16)

3 Results and Discussions

The analytical solution of the mathematical model of fluidflow in fractal reservoir was rewritten as continued fractionlike real numbers the structure of continued fraction is verybeautiful It is very convenient to study the influence of bothwellbore storage effect and skin factor using equation (15)For different outer boundary conditions the solution has adifferent kernel function but its expression is the same so thatit is convenient to program well test analysis software Boththe dimensionless formation pressure and the dimensionlesswellbore pressure in the Laplace space can be inverted backinto real time space using the Stehfest numerical inventionalgorithm (the numerical invention details are shown inAppendix B) The analysis of pressure transient response offluid flow in fractal reservoir is studied by type curves

31 Pressure Transient Response Figure 2 shows the dimen-sionless wellbore pressure and derivative responses in log-log coordinate for three kinds of outer boundary conditions(infinite constant pressure and closed) which can be dividedinto the following four seepage flow stages

1

23 4

Closed

Infinite

Transitionflow

Wellborestorage

Radial flow beforethe outer boundary Radial flow after

the outer boundary

Constant

10minus2 100 102 104 106 108 101010minus4

10minus3

10minus2

10minus1

100

101

102

103

104

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 2The type curves of the fractal reservoir with spherical flow

Stage 1 (wellbore storage period) The fluid flow is principallyaffected by wellbore storage effect In this flow period boththe dimensionless pressure change and its rate are aligned ina unite slope trend

Stage 2 (transition flow period) The period represents thetransition to early time spherical flow (Stage 3) and has ahump in the dimensionless pressure derivative curve Thefluid flow depends on both the wellbore storage effect and theskin factor

Stage 3 (spherical flow before the outer boundary period)During this period either the dimensionless pressure changecurve or the derivative curves show a straight line with a slopeof zero The greater the radius of the outer boundary is thelonger the duration of the fluid flow period will be

Stage 4 (spherical flow after the outer boundary period)When the outer boundary is infinite the fluid flow still holdsthe same response When pressure is constant in the outerboundary the dimensionless pressure curve nearly remainslevel but its derivative curve decreases quickly For a closedouter boundary condition both the dimensionless pressureand derivative curves are cocked up and overlapped And thevalues of their slope curve are about 1

As Figure 2 shows the differences of both the dimen-sionless wellbore pressure and derivative responses in var-ious outer boundary conditions only display in Stage 4For convenience we only analyze infinite outer boundarycondition The effect of reservoir parameters trends to besimilar among the three kinds of outer boundary conditions(infinite constant pressure and closed) so we analyze thecase of infinite outer boundary condition in the following

32 Effect of Wellbore Storage Figure 3 shows the effect ofthe coefficient of wellbore storage 119862119863 on pressure transientresponses of a fractal reservoir with spherical flow It can beseen that with the other parameters kept constant the curveshave obvious impact on the wellbore storage period and the

Journal of Chemistry 5

10minus4

10minus3

10minus2

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

logP

wDlo

gP998400 w

D

10minus2 100 102 104 106 108 1010

log tDCD

CD = 1 10 100 1000

Figure 3 Effect of 119862119863 for the type curves of the fractal reservoirwith spherical flow

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

10minus2 100 102 104 106 108 1010

S = 3 2 1 0

logP

wDlo

gP998400 w

D

Figure 4 Effect of 119878 for the type curves of the fractal reservoir withspherical flow

transition flow period The curves of dimensionless pressurederivative indicate that the larger coefficient of wellborestorage increases the duration of the wellbore storage period

33 Effect of Skin Factor Figure 4 shows the effect of the skinfactor 119878 on pressure transient responses of a fractal reservoirwith spherical flow If the other parameters keep constant thelarger the skin factor is the higher the hump of the wellborestorage period in the derivative becomes

34 Effect of Fractal Dimension The effect of the fractaldimension 119889119891 is shown in Figure 5 We can find thatthe fluid flow behaviors are not affected in the wellborestorage period The bigger the 119889119891 is the higher the derivativecurve will become As time goes on the difference couldbe observed obviously That is because of the fact thatwith fractal dimension increasing the fractal porosity and

10minus2 100 102 104 106 108 101010minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

df = 26 27 28 29

logP

wDlo

gP998400 w

D

Figure 5 Effect of 119889119891 for the type curves of the fractal reservoir withspherical flow

10minus2 100 102 104 106 108 101010minus2

10minus1

100

101

102

120579 = 04 05 06 07

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 6 Effect of 120579 for the type curves of the fractal reservoir withspherical flow

permeability of a reservoir will become bigger which also canbe understood from (A1)

35 Effect of Conductivity Index Figure 6 shows the influenceof conductivity index 120579 on pressure transient responses of afractal reservoir with spherical flow The conductivity indexreflects the curvature of the pore system It holds that themore circuitous pore system is the worse connectivity andconductivity will be When the fractal dimension is smallenough and makes 119889 + 120579 minus 119889119891 lt 1 the derivative curvedecreases at the late time such as the blue curve and thegreen curve because the reservoir is similar to the situationof the constant pressure outer boundary When the fractaldimensionmakes 119889+120579minus119889119891 ge 1 the derivative curve (the redcurve) inclines upward at the late time because the reservoiris similar to the situation of the closed outer boundary

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

2 Journal of Chemistry

finite-conductivity fractures [11] The papers obtained somenumerical or analytical solutions of the formation pressureand the wellbore pressure in different fractal reservoirs orplotted pressure-time curves to analyze the influence ofreservoir parameters [12ndash17]

Different versions of cylindrical flow model have beenstudied on the assumption that wells were completely openedand fully penetrated the productive formation However theassumptionmay seem too restrictive for practical applicationReservoirs have vertical permeability and the length ofinjection or extraction region was small compared to thickformations spherical flowmodel can provide a good approx-imation in practice [18] Schroth and Istok [19] illustratedthe applicability of the derived spherical flow solution andprovided a comparison with its cylindrical flow counterpartThey showed the spherical flow solution increased withincreasing anisotropy in hydraulic conductivities Joseph andKoederitz [20] presented short-time interpretation methodsfor radial-spherical flow in homogeneous and isotropic reser-voirs inclusive of wellbore storage wellbore phase redistribu-tion and damage skin effects Liu [21] studied the transientspherical flow behavior in porous media and derived its non-linear partial differential equation and obtained its analyticalasymptotic and approximate solutions by using the methodsof Laplace transform

In the above proposed studies the solution processes oftheir mathematical models are very complicated Based onsome study in a boundary value problem of the second-order linear homogeneous differential equation Li andLiao recently introduced a new idea similar constructionmethod The similar structure theory and its applicationwere developed maturely [22] Chen and Li presented similarconstruction method of solution (SCMS) for the boundaryvalue problem of Bessel equation and gave the method toanalyze the structure characteristics of solution [23] Afterthat Sheng et al proposed SCMS for a fractal reservoir andlisted its steps [17] They verified that the SCMS is a straight-forward method to solve mathematical models in reservoirengineering but they just did a theoretical study SCMS of theboundary value problems of Airy equation Weber equationEuler hypergeometric equation and composite first Webersystem were studied [24ndash28] SCMS in fractal dual-porosityreservoir and dual-permeability reservoir were presented[29ndash31]

This paper presents amathematicalmodel for the analysisof the pressure transient response of fluid spherical flowin fractal reservoir which considers wellbore storage andskin effect in inner boundary conditions According to amathematical method called SCMS the expressions of thedimensionless formation pressure and the dimensionlesswellbore pressure are constructed in the Laplace space whichavoid complicated calculations Type curves of the wellborepressure and pressure derivative responses are plotted andanalyzed in real domain using the Stehfest numerical inven-tion algorithm [32] The results obtained in this study haveessential significance to understand the pressure character-istics and provide theoretical basis in such a reservoir Thispaper embodies the relationship between engineering andmathematics

Well

Reservoir

Figure 1 Schematic for spherical flow in a reservoir

2 Materials and Methods

21 Reservoir Characteristics Nonpenetrating wells thatoccur in a thick formation can be treated as spherical systemsas shown in Figure 1 Using fractal theory to describe the porenature and transport properties we assume that dimensionalfractal network 119889119891 is embedded in the dimensional Euclideanrock 119889 The fluid only flows in the fractal network andthe flow will obey Darcyrsquos law from the reservoir into thewellbore The reservoir has a uniform thickness of ℎ andoriginal formation pressure is 119901119894 The fractal reservoir ismined with a single small opening hole in the top part Thefluid flow rate is 119902(119905) The gravity and the capillary pressureare ignored

22 Dimensionless Mathematical Model The dimensionlessmathematical model of the reservoir is made up of four partssuch as continuity equation initial condition inner bound-ary condition and outer boundary condition Appendix Apresents mathematical model of fluid flow and dimensionlessvariables

The continuity equation of fluid spherical flow in a fractalreservoir is

1205972119901119863

1205971199032119863

+120573

119903119863

120597119901119863

120597119903119863

= 119903120579119863

120597119901119863

120597119905119863

(1)

Initial condition

119901119863 (119903119863 0) = 0 (2)

Journal of Chemistry 3

Inner boundary condition

(1199032119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

= minus 119902119863 (119905119863) + 119862119863

119889119901119908119863

119889119905119863

119901119908119863 (119905119863) = (119901119863 minus 119878119903119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

(3)

Three kinds of outer boundary conditions are the follow-ing cases

Case 1 Infinite outer boundary condition

119901119863 (infin 119905119863) = 0 (4)

Case 2 Constant pressure outer boundary condition

119901119863 (119877119863 119905119863) = 0 (5)

Case 3 Closed outer boundary condition120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0 (6)

23 A Boundary Value Problem of the Modified Bessel Equa-tion The dimensionless mathematical model ((1)ndash(6)) hap-pens to be a boundary value problem of a second-orderpartial differential equation It can be transformed into aboundary value problem of a second-order ordinary differ-ential equation based on Laplace transform and then it canbe further simplified as a boundary value problem of themodified Bessel equation

1199032119863

1198892119901119863

1198891199032119863

+ 120573119903119863

119889119901119863

119889119903119863

minus 119903120579+2119863 119911119901119863 = 0 (7)

[minus119862119863119911119901119908119863 (119903119863 119911) + (1 + 119862119863119911119878)119889119901119863 (119903119863 119911)

119889119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= minus119902119863 (119911)

(8)

119901119863 (infin 119911) = 0 or 119901119863 (119877119863 119911) = 0 or119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(9)

24 Solution The SCMS is based on a boundary valueproblem of a second-order linear differential equation and isfirstly proposed by Sheng et al [17] Using coefficients of itsright and left boundary condition and its two linearly inde-pendent solutions of the differential equation the solutionof the boundary value problem can be obtained Accordingto the steps of SCMS the solution of the dimensionlessmathematical model of fractal spherical flow reservoir can beconstructed The details are as follows

Step 1 Solve the linearly independent solutions of (7)

119903(1minus120573)2119863 119868] (

2radic119911

2 + 120579119903(2+120579)2119863 )

119903(1minus120573)2119863 119870] (

2radic119911

2 + 120579119903(2+120579)2119863 )

(10)

where 119868](sdot) and 119896V(sdot) denote the first and the secondmodifiedBessel function of order ] = (1 minus 120573)(2 + 120579) respectively

Step 2 Construct a binary function 12059300(1199031198631 1199031198632) employingthe linearly independent solutions (10) and calculate itspartial derivatives for 1199031198631 1199031198632 respectively

12059300 (1199031198631 1199031198632) = (11990311986311199031198632)(1minus120573)2

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(11)

where

120595119898119899 (119909120576 119910120576 120585) = 119870119898 (120585119909

120576) 119868119899 (120585119910

120576)

+ (minus1)119898minus119899+1

119868119898 (120585119909120576)119870119899 (120585119910

120576)

(12)

with real numbers119898 119899

12059301 (1199031198631 1199031198632) =120597

1205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903(1minus120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ radic119911119903(1minus120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059310 (1199031198631 1199031198632) =120597

1205971199031198631

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903minus(1minus120573)21198631 119903

(1minus120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus radic119911119903(1minus120573+120579)21198631 119903

(1minus120573)21198632

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059311 (1199031198631 1199031198632) =1205972

12059711990311986311205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573)2119903minus(1+120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ (1 minus 120573)radic119911119903minus(1+120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus (1 minus 120573)radic119911119903(1minus120573+120579)21198631 119903

minus(1+120573)21198632

4 Journal of Chemistry

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus 119911119903(1minus120573+120579)21198631 119903

(1minus120573+120579)21198632

times 120595]+1]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(13)

Step 3 Construct similar kernel functionsΦ(119903119863 119911) using theouter boundary conditions (9) the binary function equation(11) and its partial derivatives equations (13)

Φ(119903119863 119911) =

12059300 (119903119863infin)

12059310 (1infin) 119901119863 (infin 119911) = 0

12059300 (119903119863 119877119863)

12059310 (1 119877119863) 119901119863 (119877119863 119911) = 0

12059301 (119903119863 119877119863)

12059311 (1 119877119863)

119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(14)

Step 4 Construct the similar structure of solution withthe similar kernel function equation (14) and coefficients(119902119863(119911) 119862119863119911 119878) of the inner boundary condition equation (8)

119901119863 (119903119863 119911) = 119902119863 (119911)1

119862119863119911 + (1 (119878 + Φ (1 119911)))

sdot1

119878 + Φ (1 119911)sdot Φ (119903119863 119911)

(15)

Equation (15) is the analytical solution of the mathemati-cal model of fluid spherical flow in a fractal reservoir Substi-tute (15) into the second expression of (3) the dimensionlesswellbore pressure in the Laplace space can be obtained

119901119908119863 (119911) = 119902119863 (119911)1

119862119863119911 + Φ (1 119911) (16)

3 Results and Discussions

The analytical solution of the mathematical model of fluidflow in fractal reservoir was rewritten as continued fractionlike real numbers the structure of continued fraction is verybeautiful It is very convenient to study the influence of bothwellbore storage effect and skin factor using equation (15)For different outer boundary conditions the solution has adifferent kernel function but its expression is the same so thatit is convenient to program well test analysis software Boththe dimensionless formation pressure and the dimensionlesswellbore pressure in the Laplace space can be inverted backinto real time space using the Stehfest numerical inventionalgorithm (the numerical invention details are shown inAppendix B) The analysis of pressure transient response offluid flow in fractal reservoir is studied by type curves

31 Pressure Transient Response Figure 2 shows the dimen-sionless wellbore pressure and derivative responses in log-log coordinate for three kinds of outer boundary conditions(infinite constant pressure and closed) which can be dividedinto the following four seepage flow stages

1

23 4

Closed

Infinite

Transitionflow

Wellborestorage

Radial flow beforethe outer boundary Radial flow after

the outer boundary

Constant

10minus2 100 102 104 106 108 101010minus4

10minus3

10minus2

10minus1

100

101

102

103

104

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 2The type curves of the fractal reservoir with spherical flow

Stage 1 (wellbore storage period) The fluid flow is principallyaffected by wellbore storage effect In this flow period boththe dimensionless pressure change and its rate are aligned ina unite slope trend

Stage 2 (transition flow period) The period represents thetransition to early time spherical flow (Stage 3) and has ahump in the dimensionless pressure derivative curve Thefluid flow depends on both the wellbore storage effect and theskin factor

Stage 3 (spherical flow before the outer boundary period)During this period either the dimensionless pressure changecurve or the derivative curves show a straight line with a slopeof zero The greater the radius of the outer boundary is thelonger the duration of the fluid flow period will be

Stage 4 (spherical flow after the outer boundary period)When the outer boundary is infinite the fluid flow still holdsthe same response When pressure is constant in the outerboundary the dimensionless pressure curve nearly remainslevel but its derivative curve decreases quickly For a closedouter boundary condition both the dimensionless pressureand derivative curves are cocked up and overlapped And thevalues of their slope curve are about 1

As Figure 2 shows the differences of both the dimen-sionless wellbore pressure and derivative responses in var-ious outer boundary conditions only display in Stage 4For convenience we only analyze infinite outer boundarycondition The effect of reservoir parameters trends to besimilar among the three kinds of outer boundary conditions(infinite constant pressure and closed) so we analyze thecase of infinite outer boundary condition in the following

32 Effect of Wellbore Storage Figure 3 shows the effect ofthe coefficient of wellbore storage 119862119863 on pressure transientresponses of a fractal reservoir with spherical flow It can beseen that with the other parameters kept constant the curveshave obvious impact on the wellbore storage period and the

Journal of Chemistry 5

10minus4

10minus3

10minus2

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

logP

wDlo

gP998400 w

D

10minus2 100 102 104 106 108 1010

log tDCD

CD = 1 10 100 1000

Figure 3 Effect of 119862119863 for the type curves of the fractal reservoirwith spherical flow

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

10minus2 100 102 104 106 108 1010

S = 3 2 1 0

logP

wDlo

gP998400 w

D

Figure 4 Effect of 119878 for the type curves of the fractal reservoir withspherical flow

transition flow period The curves of dimensionless pressurederivative indicate that the larger coefficient of wellborestorage increases the duration of the wellbore storage period

33 Effect of Skin Factor Figure 4 shows the effect of the skinfactor 119878 on pressure transient responses of a fractal reservoirwith spherical flow If the other parameters keep constant thelarger the skin factor is the higher the hump of the wellborestorage period in the derivative becomes

34 Effect of Fractal Dimension The effect of the fractaldimension 119889119891 is shown in Figure 5 We can find thatthe fluid flow behaviors are not affected in the wellborestorage period The bigger the 119889119891 is the higher the derivativecurve will become As time goes on the difference couldbe observed obviously That is because of the fact thatwith fractal dimension increasing the fractal porosity and

10minus2 100 102 104 106 108 101010minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

df = 26 27 28 29

logP

wDlo

gP998400 w

D

Figure 5 Effect of 119889119891 for the type curves of the fractal reservoir withspherical flow

10minus2 100 102 104 106 108 101010minus2

10minus1

100

101

102

120579 = 04 05 06 07

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 6 Effect of 120579 for the type curves of the fractal reservoir withspherical flow

permeability of a reservoir will become bigger which also canbe understood from (A1)

35 Effect of Conductivity Index Figure 6 shows the influenceof conductivity index 120579 on pressure transient responses of afractal reservoir with spherical flow The conductivity indexreflects the curvature of the pore system It holds that themore circuitous pore system is the worse connectivity andconductivity will be When the fractal dimension is smallenough and makes 119889 + 120579 minus 119889119891 lt 1 the derivative curvedecreases at the late time such as the blue curve and thegreen curve because the reservoir is similar to the situationof the constant pressure outer boundary When the fractaldimensionmakes 119889+120579minus119889119891 ge 1 the derivative curve (the redcurve) inclines upward at the late time because the reservoiris similar to the situation of the closed outer boundary

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Journal of Chemistry 3

Inner boundary condition

(1199032119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

= minus 119902119863 (119905119863) + 119862119863

119889119901119908119863

119889119905119863

119901119908119863 (119905119863) = (119901119863 minus 119878119903119863

120597119901119863

120597119903119863

)

10038161003816100381610038161003816100381610038161003816119903119863=1

(3)

Three kinds of outer boundary conditions are the follow-ing cases

Case 1 Infinite outer boundary condition

119901119863 (infin 119905119863) = 0 (4)

Case 2 Constant pressure outer boundary condition

119901119863 (119877119863 119905119863) = 0 (5)

Case 3 Closed outer boundary condition120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0 (6)

23 A Boundary Value Problem of the Modified Bessel Equa-tion The dimensionless mathematical model ((1)ndash(6)) hap-pens to be a boundary value problem of a second-orderpartial differential equation It can be transformed into aboundary value problem of a second-order ordinary differ-ential equation based on Laplace transform and then it canbe further simplified as a boundary value problem of themodified Bessel equation

1199032119863

1198892119901119863

1198891199032119863

+ 120573119903119863

119889119901119863

119889119903119863

minus 119903120579+2119863 119911119901119863 = 0 (7)

[minus119862119863119911119901119908119863 (119903119863 119911) + (1 + 119862119863119911119878)119889119901119863 (119903119863 119911)

119889119903119863

]

100381610038161003816100381610038161003816100381610038161003816119903119863=1

= minus119902119863 (119911)

(8)

119901119863 (infin 119911) = 0 or 119901119863 (119877119863 119911) = 0 or119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(9)

24 Solution The SCMS is based on a boundary valueproblem of a second-order linear differential equation and isfirstly proposed by Sheng et al [17] Using coefficients of itsright and left boundary condition and its two linearly inde-pendent solutions of the differential equation the solutionof the boundary value problem can be obtained Accordingto the steps of SCMS the solution of the dimensionlessmathematical model of fractal spherical flow reservoir can beconstructed The details are as follows

Step 1 Solve the linearly independent solutions of (7)

119903(1minus120573)2119863 119868] (

2radic119911

2 + 120579119903(2+120579)2119863 )

119903(1minus120573)2119863 119870] (

2radic119911

2 + 120579119903(2+120579)2119863 )

(10)

where 119868](sdot) and 119896V(sdot) denote the first and the secondmodifiedBessel function of order ] = (1 minus 120573)(2 + 120579) respectively

Step 2 Construct a binary function 12059300(1199031198631 1199031198632) employingthe linearly independent solutions (10) and calculate itspartial derivatives for 1199031198631 1199031198632 respectively

12059300 (1199031198631 1199031198632) = (11990311986311199031198632)(1minus120573)2

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(11)

where

120595119898119899 (119909120576 119910120576 120585) = 119870119898 (120585119909

120576) 119868119899 (120585119910

120576)

+ (minus1)119898minus119899+1

119868119898 (120585119909120576)119870119899 (120585119910

120576)

(12)

with real numbers119898 119899

12059301 (1199031198631 1199031198632) =120597

1205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903(1minus120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ radic119911119903(1minus120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059310 (1199031198631 1199031198632) =120597

1205971199031198631

12059300 (1199031198631 1199031198632)

= (1 minus 120573) 119903minus(1minus120573)21198631 119903

(1minus120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus radic119911119903(1minus120573+120579)21198631 119903

(1minus120573)21198632

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

12059311 (1199031198631 1199031198632) =1205972

12059711990311986311205971199031198632

12059300 (1199031198631 1199031198632)

= (1 minus 120573)2119903minus(1+120573)21198631 119903

minus(1+120573)21198632

times 120595]] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

+ (1 minus 120573)radic119911119903minus(1+120573)21198631 119903

(1minus120573+120579)21198632

times 120595]]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus (1 minus 120573)radic119911119903(1minus120573+120579)21198631 119903

minus(1+120573)21198632

4 Journal of Chemistry

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus 119911119903(1minus120573+120579)21198631 119903

(1minus120573+120579)21198632

times 120595]+1]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(13)

Step 3 Construct similar kernel functionsΦ(119903119863 119911) using theouter boundary conditions (9) the binary function equation(11) and its partial derivatives equations (13)

Φ(119903119863 119911) =

12059300 (119903119863infin)

12059310 (1infin) 119901119863 (infin 119911) = 0

12059300 (119903119863 119877119863)

12059310 (1 119877119863) 119901119863 (119877119863 119911) = 0

12059301 (119903119863 119877119863)

12059311 (1 119877119863)

119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(14)

Step 4 Construct the similar structure of solution withthe similar kernel function equation (14) and coefficients(119902119863(119911) 119862119863119911 119878) of the inner boundary condition equation (8)

119901119863 (119903119863 119911) = 119902119863 (119911)1

119862119863119911 + (1 (119878 + Φ (1 119911)))

sdot1

119878 + Φ (1 119911)sdot Φ (119903119863 119911)

(15)

Equation (15) is the analytical solution of the mathemati-cal model of fluid spherical flow in a fractal reservoir Substi-tute (15) into the second expression of (3) the dimensionlesswellbore pressure in the Laplace space can be obtained

119901119908119863 (119911) = 119902119863 (119911)1

119862119863119911 + Φ (1 119911) (16)

3 Results and Discussions

The analytical solution of the mathematical model of fluidflow in fractal reservoir was rewritten as continued fractionlike real numbers the structure of continued fraction is verybeautiful It is very convenient to study the influence of bothwellbore storage effect and skin factor using equation (15)For different outer boundary conditions the solution has adifferent kernel function but its expression is the same so thatit is convenient to program well test analysis software Boththe dimensionless formation pressure and the dimensionlesswellbore pressure in the Laplace space can be inverted backinto real time space using the Stehfest numerical inventionalgorithm (the numerical invention details are shown inAppendix B) The analysis of pressure transient response offluid flow in fractal reservoir is studied by type curves

31 Pressure Transient Response Figure 2 shows the dimen-sionless wellbore pressure and derivative responses in log-log coordinate for three kinds of outer boundary conditions(infinite constant pressure and closed) which can be dividedinto the following four seepage flow stages

1

23 4

Closed

Infinite

Transitionflow

Wellborestorage

Radial flow beforethe outer boundary Radial flow after

the outer boundary

Constant

10minus2 100 102 104 106 108 101010minus4

10minus3

10minus2

10minus1

100

101

102

103

104

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 2The type curves of the fractal reservoir with spherical flow

Stage 1 (wellbore storage period) The fluid flow is principallyaffected by wellbore storage effect In this flow period boththe dimensionless pressure change and its rate are aligned ina unite slope trend

Stage 2 (transition flow period) The period represents thetransition to early time spherical flow (Stage 3) and has ahump in the dimensionless pressure derivative curve Thefluid flow depends on both the wellbore storage effect and theskin factor

Stage 3 (spherical flow before the outer boundary period)During this period either the dimensionless pressure changecurve or the derivative curves show a straight line with a slopeof zero The greater the radius of the outer boundary is thelonger the duration of the fluid flow period will be

Stage 4 (spherical flow after the outer boundary period)When the outer boundary is infinite the fluid flow still holdsthe same response When pressure is constant in the outerboundary the dimensionless pressure curve nearly remainslevel but its derivative curve decreases quickly For a closedouter boundary condition both the dimensionless pressureand derivative curves are cocked up and overlapped And thevalues of their slope curve are about 1

As Figure 2 shows the differences of both the dimen-sionless wellbore pressure and derivative responses in var-ious outer boundary conditions only display in Stage 4For convenience we only analyze infinite outer boundarycondition The effect of reservoir parameters trends to besimilar among the three kinds of outer boundary conditions(infinite constant pressure and closed) so we analyze thecase of infinite outer boundary condition in the following

32 Effect of Wellbore Storage Figure 3 shows the effect ofthe coefficient of wellbore storage 119862119863 on pressure transientresponses of a fractal reservoir with spherical flow It can beseen that with the other parameters kept constant the curveshave obvious impact on the wellbore storage period and the

Journal of Chemistry 5

10minus4

10minus3

10minus2

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

logP

wDlo

gP998400 w

D

10minus2 100 102 104 106 108 1010

log tDCD

CD = 1 10 100 1000

Figure 3 Effect of 119862119863 for the type curves of the fractal reservoirwith spherical flow

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

10minus2 100 102 104 106 108 1010

S = 3 2 1 0

logP

wDlo

gP998400 w

D

Figure 4 Effect of 119878 for the type curves of the fractal reservoir withspherical flow

transition flow period The curves of dimensionless pressurederivative indicate that the larger coefficient of wellborestorage increases the duration of the wellbore storage period

33 Effect of Skin Factor Figure 4 shows the effect of the skinfactor 119878 on pressure transient responses of a fractal reservoirwith spherical flow If the other parameters keep constant thelarger the skin factor is the higher the hump of the wellborestorage period in the derivative becomes

34 Effect of Fractal Dimension The effect of the fractaldimension 119889119891 is shown in Figure 5 We can find thatthe fluid flow behaviors are not affected in the wellborestorage period The bigger the 119889119891 is the higher the derivativecurve will become As time goes on the difference couldbe observed obviously That is because of the fact thatwith fractal dimension increasing the fractal porosity and

10minus2 100 102 104 106 108 101010minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

df = 26 27 28 29

logP

wDlo

gP998400 w

D

Figure 5 Effect of 119889119891 for the type curves of the fractal reservoir withspherical flow

10minus2 100 102 104 106 108 101010minus2

10minus1

100

101

102

120579 = 04 05 06 07

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 6 Effect of 120579 for the type curves of the fractal reservoir withspherical flow

permeability of a reservoir will become bigger which also canbe understood from (A1)

35 Effect of Conductivity Index Figure 6 shows the influenceof conductivity index 120579 on pressure transient responses of afractal reservoir with spherical flow The conductivity indexreflects the curvature of the pore system It holds that themore circuitous pore system is the worse connectivity andconductivity will be When the fractal dimension is smallenough and makes 119889 + 120579 minus 119889119891 lt 1 the derivative curvedecreases at the late time such as the blue curve and thegreen curve because the reservoir is similar to the situationof the constant pressure outer boundary When the fractaldimensionmakes 119889+120579minus119889119891 ge 1 the derivative curve (the redcurve) inclines upward at the late time because the reservoiris similar to the situation of the closed outer boundary

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

4 Journal of Chemistry

times 120595]+1] (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

minus 119911119903(1minus120573+120579)21198631 119903

(1minus120573+120579)21198632

times 120595]+1]+1 (119903(2+120579)21198631 119903

(2+120579)21198632

2radic119911

2 + 120579)

(13)

Step 3 Construct similar kernel functionsΦ(119903119863 119911) using theouter boundary conditions (9) the binary function equation(11) and its partial derivatives equations (13)

Φ(119903119863 119911) =

12059300 (119903119863infin)

12059310 (1infin) 119901119863 (infin 119911) = 0

12059300 (119903119863 119877119863)

12059310 (1 119877119863) 119901119863 (119877119863 119911) = 0

12059301 (119903119863 119877119863)

12059311 (1 119877119863)

119889119901119863

119889119903119863

100381610038161003816100381610038161003816100381610038161003816119903119863=119877119863

= 0

(14)

Step 4 Construct the similar structure of solution withthe similar kernel function equation (14) and coefficients(119902119863(119911) 119862119863119911 119878) of the inner boundary condition equation (8)

119901119863 (119903119863 119911) = 119902119863 (119911)1

119862119863119911 + (1 (119878 + Φ (1 119911)))

sdot1

119878 + Φ (1 119911)sdot Φ (119903119863 119911)

(15)

Equation (15) is the analytical solution of the mathemati-cal model of fluid spherical flow in a fractal reservoir Substi-tute (15) into the second expression of (3) the dimensionlesswellbore pressure in the Laplace space can be obtained

119901119908119863 (119911) = 119902119863 (119911)1

119862119863119911 + Φ (1 119911) (16)

3 Results and Discussions

The analytical solution of the mathematical model of fluidflow in fractal reservoir was rewritten as continued fractionlike real numbers the structure of continued fraction is verybeautiful It is very convenient to study the influence of bothwellbore storage effect and skin factor using equation (15)For different outer boundary conditions the solution has adifferent kernel function but its expression is the same so thatit is convenient to program well test analysis software Boththe dimensionless formation pressure and the dimensionlesswellbore pressure in the Laplace space can be inverted backinto real time space using the Stehfest numerical inventionalgorithm (the numerical invention details are shown inAppendix B) The analysis of pressure transient response offluid flow in fractal reservoir is studied by type curves

31 Pressure Transient Response Figure 2 shows the dimen-sionless wellbore pressure and derivative responses in log-log coordinate for three kinds of outer boundary conditions(infinite constant pressure and closed) which can be dividedinto the following four seepage flow stages

1

23 4

Closed

Infinite

Transitionflow

Wellborestorage

Radial flow beforethe outer boundary Radial flow after

the outer boundary

Constant

10minus2 100 102 104 106 108 101010minus4

10minus3

10minus2

10minus1

100

101

102

103

104

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 2The type curves of the fractal reservoir with spherical flow

Stage 1 (wellbore storage period) The fluid flow is principallyaffected by wellbore storage effect In this flow period boththe dimensionless pressure change and its rate are aligned ina unite slope trend

Stage 2 (transition flow period) The period represents thetransition to early time spherical flow (Stage 3) and has ahump in the dimensionless pressure derivative curve Thefluid flow depends on both the wellbore storage effect and theskin factor

Stage 3 (spherical flow before the outer boundary period)During this period either the dimensionless pressure changecurve or the derivative curves show a straight line with a slopeof zero The greater the radius of the outer boundary is thelonger the duration of the fluid flow period will be

Stage 4 (spherical flow after the outer boundary period)When the outer boundary is infinite the fluid flow still holdsthe same response When pressure is constant in the outerboundary the dimensionless pressure curve nearly remainslevel but its derivative curve decreases quickly For a closedouter boundary condition both the dimensionless pressureand derivative curves are cocked up and overlapped And thevalues of their slope curve are about 1

As Figure 2 shows the differences of both the dimen-sionless wellbore pressure and derivative responses in var-ious outer boundary conditions only display in Stage 4For convenience we only analyze infinite outer boundarycondition The effect of reservoir parameters trends to besimilar among the three kinds of outer boundary conditions(infinite constant pressure and closed) so we analyze thecase of infinite outer boundary condition in the following

32 Effect of Wellbore Storage Figure 3 shows the effect ofthe coefficient of wellbore storage 119862119863 on pressure transientresponses of a fractal reservoir with spherical flow It can beseen that with the other parameters kept constant the curveshave obvious impact on the wellbore storage period and the

Journal of Chemistry 5

10minus4

10minus3

10minus2

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

logP

wDlo

gP998400 w

D

10minus2 100 102 104 106 108 1010

log tDCD

CD = 1 10 100 1000

Figure 3 Effect of 119862119863 for the type curves of the fractal reservoirwith spherical flow

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

10minus2 100 102 104 106 108 1010

S = 3 2 1 0

logP

wDlo

gP998400 w

D

Figure 4 Effect of 119878 for the type curves of the fractal reservoir withspherical flow

transition flow period The curves of dimensionless pressurederivative indicate that the larger coefficient of wellborestorage increases the duration of the wellbore storage period

33 Effect of Skin Factor Figure 4 shows the effect of the skinfactor 119878 on pressure transient responses of a fractal reservoirwith spherical flow If the other parameters keep constant thelarger the skin factor is the higher the hump of the wellborestorage period in the derivative becomes

34 Effect of Fractal Dimension The effect of the fractaldimension 119889119891 is shown in Figure 5 We can find thatthe fluid flow behaviors are not affected in the wellborestorage period The bigger the 119889119891 is the higher the derivativecurve will become As time goes on the difference couldbe observed obviously That is because of the fact thatwith fractal dimension increasing the fractal porosity and

10minus2 100 102 104 106 108 101010minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

df = 26 27 28 29

logP

wDlo

gP998400 w

D

Figure 5 Effect of 119889119891 for the type curves of the fractal reservoir withspherical flow

10minus2 100 102 104 106 108 101010minus2

10minus1

100

101

102

120579 = 04 05 06 07

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 6 Effect of 120579 for the type curves of the fractal reservoir withspherical flow

permeability of a reservoir will become bigger which also canbe understood from (A1)

35 Effect of Conductivity Index Figure 6 shows the influenceof conductivity index 120579 on pressure transient responses of afractal reservoir with spherical flow The conductivity indexreflects the curvature of the pore system It holds that themore circuitous pore system is the worse connectivity andconductivity will be When the fractal dimension is smallenough and makes 119889 + 120579 minus 119889119891 lt 1 the derivative curvedecreases at the late time such as the blue curve and thegreen curve because the reservoir is similar to the situationof the constant pressure outer boundary When the fractaldimensionmakes 119889+120579minus119889119891 ge 1 the derivative curve (the redcurve) inclines upward at the late time because the reservoiris similar to the situation of the closed outer boundary

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Journal of Chemistry 5

10minus4

10minus3

10minus2

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

logP

wDlo

gP998400 w

D

10minus2 100 102 104 106 108 1010

log tDCD

CD = 1 10 100 1000

Figure 3 Effect of 119862119863 for the type curves of the fractal reservoirwith spherical flow

10minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

10minus2 100 102 104 106 108 1010

S = 3 2 1 0

logP

wDlo

gP998400 w

D

Figure 4 Effect of 119878 for the type curves of the fractal reservoir withspherical flow

transition flow period The curves of dimensionless pressurederivative indicate that the larger coefficient of wellborestorage increases the duration of the wellbore storage period

33 Effect of Skin Factor Figure 4 shows the effect of the skinfactor 119878 on pressure transient responses of a fractal reservoirwith spherical flow If the other parameters keep constant thelarger the skin factor is the higher the hump of the wellborestorage period in the derivative becomes

34 Effect of Fractal Dimension The effect of the fractaldimension 119889119891 is shown in Figure 5 We can find thatthe fluid flow behaviors are not affected in the wellborestorage period The bigger the 119889119891 is the higher the derivativecurve will become As time goes on the difference couldbe observed obviously That is because of the fact thatwith fractal dimension increasing the fractal porosity and

10minus2 100 102 104 106 108 101010minus1

100

101

102

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

df = 26 27 28 29

logP

wDlo

gP998400 w

D

Figure 5 Effect of 119889119891 for the type curves of the fractal reservoir withspherical flow

10minus2 100 102 104 106 108 101010minus2

10minus1

100

101

102

120579 = 04 05 06 07

Dim

ensio

nles

s pre

ssur

e and

der

ivat

ive

log tDCD

logP

wDlo

gP998400 w

D

Figure 6 Effect of 120579 for the type curves of the fractal reservoir withspherical flow

permeability of a reservoir will become bigger which also canbe understood from (A1)

35 Effect of Conductivity Index Figure 6 shows the influenceof conductivity index 120579 on pressure transient responses of afractal reservoir with spherical flow The conductivity indexreflects the curvature of the pore system It holds that themore circuitous pore system is the worse connectivity andconductivity will be When the fractal dimension is smallenough and makes 119889 + 120579 minus 119889119891 lt 1 the derivative curvedecreases at the late time such as the blue curve and thegreen curve because the reservoir is similar to the situationof the constant pressure outer boundary When the fractaldimensionmakes 119889+120579minus119889119891 ge 1 the derivative curve (the redcurve) inclines upward at the late time because the reservoiris similar to the situation of the closed outer boundary

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

6 Journal of Chemistry

4 Conclusions

Based on the reasoning mentioned above the followingconclusions can be drawn

(1) The mathematical models of fluid spherical flow infractal reservoirs with three kinds of outer boundaryconditions (infinite constant pressure and closed)were presented which comprehensively took intoconsideration the effect of wellbore storage effect theskin factor fractal dimension and conductivity index

(2) The dimensionless formation pressure in the Laplacespace was constructed by SCMS and was written as acontinued fraction The analytical solution presentedin this paper has afforded theoretical basis to under-stand the effect of reservoir parameters and practicalsignificance to make the well test analysis software

(3) The dimensionless pressure and derivative typecurves were plotted and their characteristics wereanalyzed For three kinds of outer boundary con-ditions the pressure transient responses make nodifference when the fluid does not reach to the outerboundary condition

(4) The time and the high of the hump of the dimen-sionless wellbore pressure derivative have connectionwith the coefficient of wellbore storage 119862119863 and theskin factor 119878 respectively The paper has studied theeffects of the conductivity index 120579 and the fractaldimension 119889119891 The fractal dimension 119889119891 is beneficialto the fluid flow However the trend is reversing withthe conductivity index 120579 In addition the behaviors inwellbore storage period are not affected for the fractaldimension 119889119891 and the conductivity index 120579

Appendices

A Mathematical Model of Fractal SphericalFlow Reservoir

In Acuna et alrsquos research [2] the fractal porosity and perme-ability of a reservoir as a function of distance are

120601 (119903) = 120601119908 (119903

119903119908

)

119889119891minus119889

119896 (119903) = 119896119908 (119903

119903119908

)

119889119891minus119889minus120579

(A1)

where 119889 119889119891 120579 and 119903 represent Euclid dimension fractaldimension conductivity index and radial distance respec-tively

The fluid flow from reservoir into the wellbore followsDarcyrsquos law The fluid flow in the reservoir is isothermaland single-phase with constant fluid viscosity Based on theprinciple of mass and energy conservation the mathematicalequations of single fluid phase in fractal reservoir can bewritten in the form as

1205972119901

1205971199032+

120573

119903

120597119901

120597119903= (

119903

119903119908

)

120579120583120601119908119862119905

119896119908

120597119901

120597119905 (A2)

where

120573 = 119889119891 minus 119889 minus 120579 + 2 119862119905 = 119862119897 + 119862119891

119862119897 =1

120588

120597120588

120597119901 119862119891 =

1

120601119891

120597120601119891

120597119901

(A3)

Initial condition

119901 (119903 0) = 0 (A4)

Inner boundary conditions

(1199032 120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

=1842 times 10

minus3120583

119896119908

[119861119902 (119905) + 119862119889119901119908

119889119905]

119901119908 (119905) = (119901 (119903119908 119905) minus 119878119903120597119901

120597119903)

10038161003816100381610038161003816100381610038161003816119903=119903119908

(A5)

The three kinds of outer boundary condition are asfollows

Case 1 Infinite outer boundary condition

119901 (infin 119905) = 1199010 (A6)

Case 2 Constant pressure outer boundary condition

119901 (119877 119905) = 1199010 (A7)

Case 3 Closed outer boundary condition

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119877= 0 (A8)

For the convenience of calculation the dimensionlessvariables (formation pressure wellbore pressure productionrate radial distance outer boundary radius time and well-bore storage coefficient) are defined as follows

119901119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901)

119901119908119863 =119896119908119903119908

1842 times 10minus3119861120583119902119890

(119901119894 minus 119901119908)

119902119863 (119905) =119902 (119905)

119902119890

119903119863 =119903

119903119908

119877119863 =119877

119903119908

119905119863 =365 times 10

minus3119896119908119905

1205831206011199081198621199051199032119908

119862119863 =796 times 10

minus2119862

1206011199081198621199051199033119908

(A9)

Taking the above dimensionless variables into (A2)ndash(A8) the dimensionlessmathematical model of fractal reser-voir ((1)ndash(6)) can be obtained

B Stehfest Numerical Invention Algorithm

The solution in real space can be obtained by the method ofthe Stehfest numerical invention algorithm [32] 119901119908119863(119905119863) is

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Journal of Chemistry 7

the primary function of 119901119908119863(119911) using the Laplace transformthen 119901119908119863(119905119863) is given by Wooden et al [33] Consider

119901119908119863 (119905119863) =ln 2

119905119863

119873

sum

119894=1

119881119894119901119908119863 (ln 2

119905119863

119894) (B1)

where119881119894 = (minus1)

(1198732)+119894

times

min(1198941198732)sum

119896=[(119894+1)2]

(1198961198732

(2119896 + 1))

times ((119896 + 1)119896 ((119873

2) minus 119896 + 1)

times (119894 minus 119896 + 1) (2119896 minus 119894 + 1))

minus1

(B2)

Here 119873 is an even number which is usually in the range10 lt 119873 lt 30 and 119896 is computed using integer arithmetic

Nomenclature

119861 Formation volume factor RBSTB119862 Coefficient of wellbore storage m3Pa119862119905 Total compressibility MPaminus1119889 Euclid dimension119889119891 Fractal dimension119896 Permeability mD119901 Reservoir pressure MPa119902 Production rate or injection rate m3d119877 Radial distance of the outer boundary m119903 Radial distance in spherical coordinate m119878 Skin factor119905 Time h119911 Laplace transform variable

Greek Symbols

120579 Conductivity index120583 Viscosity mPasdots120601 Porosity

Subscripts

119863 Dimensionless119894 Initial119908 Wellbore parameter

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee forhisher helpful suggestions and comments The research is

supported by the National Natural Science Foundation ofChina (no 51274169) theNational Basic Research ProgramofChina (973 Program) (no 2013CB228004) the New CenturyExcellent Talents in University (no NCET-11-1062) and theScientific Research Fund of Sichuan Provincial EducationDepartment of China (no 12ZA164)

References

[1] B B Mandelbrot ldquoFractal geometry what is it and what doesit dordquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 423 no1864 pp 3ndash16 1989

[2] J A Acuna I Ershagh and Y C Yortsos Fractal Analysis ofPressure Transient in the Geysers Geothermal Field PetroleumEngineering Program University of Southern California LosAngeles Calif USA 1992

[3] J Cai and B Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011

[4] J C Cai X Y Hu D C Standnes and L J You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[5] J Chang andYC Yortsos ldquoPressure-transient analysis of fractalreservoirsrdquo SPE Formation Evaluation vol 5 no 1 pp 31ndash381990

[6] J Chang and Y C Yortsos ldquoNote on pressure-transient analysisof fractal reservoirsrdquo SPE Advanced Technology Series vol 1 no2 pp 170ndash171 1993

[7] J Tian and D K Tong ldquoThe flow analysis of fluids in fractalreservoir with the fractional derivativerdquo Journal of Hydrody-namics B vol 18 no 3 pp 287ndash293 2006

[8] P Xu and B Yu ldquoDeveloping a new form of permeability andKozeny-Carman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[9] X Peng B Yu X Qiao S Qiu and Z Jiang ldquoRadial perme-ability of fractured porous media by Monte Carlo simulationsrdquoInternational Journal of Heat and Mass Transfer vol 57 no 1pp 369ndash374 2013

[10] P Xu S Qiu B Yu and Z Jiang ldquoPrediction of relative perme-ability in unsaturated porous media with a fractal approachrdquoInternational Journal of Heat and Mass Transfer vol 64 pp829ndash837 2013

[11] M C Santizo ldquoA semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theoryrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash18 2012163057-STU

[12] M A Prott and W C Chin ldquoNew exact spherical flowsolution with storage and skin for early-time interpretationwith applications to wireline formation and early-evaluationdrillstem testingrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-49140-MS pp 463ndash478 NewOrleans La USA September 1998

[13] B Q He and K L Xiang ldquoAnalytical solutions of effective wellradius model of unsteady flow in fractal reservoirsrdquo Journal ofSouthwest Petroleum Institute vol 22 no 3 pp 37ndash40 2000(Chinese)

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 Journal of Chemistry

[14] F Flamenco-Lopez and R Camacho-Velazquez ldquoDetermina-tion of fractal parameters of fracture networks using pressure-transient datardquo SPE Reservoir Evaluation and Engineering vol6 no 1 pp 39ndash47 2003

[15] X KongD Li andD Lu ldquoTransient pressure analysis in porousand fractured fractal reservoirsrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2700ndash2708 2009

[16] C C Sheng S C Li and D X Xu ldquoFractal sphere seepageflow oil-gas reservoir model and its solutionrdquo Oil and Gas FieldDevelopment vol 29 no 6 pp 55ndash58 2011 (Chinese)

[17] C C Sheng J Z Zhao Y M Li S C Li and H Jia ldquoSimilarconstruction method of solution for solving the mathematicalmodel of fractal reservoirwith spherical flowrdquo Journal of AppliedMathematics vol 2013 Article ID 219218 8 pages 2013

[18] R W Tang ldquoModel of limited-entry completions undergoingspherical flowrdquo SPE Formation Evaluation vol 3 no 4 pp 761ndash770 1988

[19] M H Schroth and J D Istok ldquoApproximate solution for solutetransport during spherical-flow push-pull testsrdquoGroundWatervol 43 no 2 pp 280ndash284 2005

[20] J A Joseph and L F Koederitz ldquoUnsteady-state spherical flowwith storage and skinrdquo Society of Petroleum Engineers journalvol 25 no 6 pp 804ndash822 1985

[21] C Q Liu ldquoTransient spherical flow of non-newtonian power-law fluids in porous mediardquo Applied Mathematics and Mechan-ics vol 9 no 6 pp 521ndash525 1988

[22] S C Li and Z J Liao ldquoConstructing the solution to a boundaryvalue problem for a differential equation with its arbitrary non-trivial solutionrdquo Journal of Sichuan University vol 49 no 6 pp1209ndash1213 2012 (Chinese)

[23] Z R Chen and S C Li ldquoThe similar structure method solvingthe boundary value problem of Bessel equationrdquo Journal ofSichuan Normal University vol 34 no 6 pp 850ndash853 2011(Chinese)

[24] F R Wang S C Li and D X Xu ldquoThe similarity constructionmethod of a kind of boundary-value problem of Airy equationrdquoJournal of Hubei NormalUniversity vol 33 no 1 pp 79ndash85 2013(Chinese)

[25] R J Huang S C Li and D X Xu ldquoSimilar structured methodof solution to boundary value of first Weber equationrdquo Journalof Mianyang Normal University vol 31 no 11 pp 1ndash5 2012(Chinese)

[26] X X Xiao and S C Li ldquoSimilar structure of solutions to aboundary value problem for Eulerrsquos hypergeometric differentialequationrdquo Journal of Inner Mongolia Normal University vol 41no 6 pp 597ndash600 2012 (Chinese)

[27] J H Shi S C Li X L Wang and D D Gui ldquoA newmethod for solving boundary value problem of compositeHermit Equationsrdquo Advanced Materials Research vol 753ndash755pp 2851ndash2854 2013

[28] X X Dong S C Li D D Gui J Pu and H C Li ldquoSimilarconstructing method for solving the boundary value problemof the composite first Weber systemrdquo The American Journal ofApplied Mathematics and Statistics vol 1 no 4 pp 76ndash82 2013

[29] X T Bao S C Li and D D Gui ldquoSimilar constructive methodfor solving the nonlinear spherical percolation model in dual-porosity mediardquo Advanced Materials Research vol 631-632 pp265ndash271 2013

[30] L Xu X Liu L Liang S Li and L Zhou ldquoThe similar structuremethod for solving themodel of fractal dual-porosity reservoirrdquoMathematical Problems in Engineering vol 2013 Article ID954106 9 pages 2013

[31] C Y Fan S C Li D D Gui M Hu and H C Li ldquoSimilarconstructing method for solving the model of the plane radialflow of dual permeability reservoirrdquo Applied Mechanics andMaterials vol 419 pp 43ndash50 2013

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

[33] B Wooden M Azari and M Soliman ldquoWell test analysisbenefits from new method of Laplace space inversionrdquo Journalof Oil amp Gas vol 90 no 29 pp 108ndash110 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of