Research ArticleA New Way to Calculate Flow Pressure for Low Permeability OilWell with Partially Penetrating Fracture

Xiong Ping12 Liu Hailong 34 Hu Haixia2 andWang Guan5

1College of Earth Sciences Yangtze University Wuhan 430100 China2College of Engineering and Technology Yangtze University Jingzhou 434020 China3China Petrochemical Exploration and Development Research Institute Beijing 100083 China4Key Laboratory of Marine Oil amp Gas Reservoirs Production Sinopec Beijing 100083 China5Department of Geosciences The University of Tulsa Tulsa OK 74104 USA

Correspondence should be addressed to Liu Hailong 478277608qqcom

Received 25 May 2017 Revised 29 August 2017 Accepted 21 March 2018 Published 30 May 2018

Academic Editor Luca Heltai

Copyright copy 2018 Xiong Ping et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to improve the validity of the previous models on calculating flow pressure for oil well with partially perforatingfracture a new physical model that obeys the actual heterogeneous reservoir characteristics was built Different conditionsincluding reservoir with impermeable top and bottom borders or the reservoir top which has constant pressure were consideredThrough dimensionless transformation Laplace transformation Fourier cosine transformation separation of variables and othermathematical methods the analytical solution of Laplace domain was obtained By using Stephenson numerical methods thenumerical solution pressure in a real domain was obtained The results of this method agree with the numerical simulationssuggesting that this newmethod is reliableThe following sensitivity analysis showed that the pressure dynamic linear flow curve canbe divided into four flow streams of early linear flow midradial flow advanced spherical flow and border controlling flow Fracturelength controls the early linear flow Permeability anisotropy significantly affects the midradial flowThe degree of penetration andfracture orientation dominantly affect the late spherical flowThe boundary conditions and reservoir boundary width mainly affectthe border controlling flow The method can be used to determine the optimal degree of opening shot vertical permeability andother useful parameters providing theoretical guidance for reservoir engineering analysis

1 Introduction

Low permeability carbonate reservoirs generally have thecharacteristics of large thickness and natural fracturesIt is well known that the perforation completion is usedin well completion and the hydraulic fracturing is usedto increase the well production However the degree ofperforation is relatively small (a few meters) which canaffect the wellborersquos pressure transmission and productionwellsrsquo productivity [1ndash3] Therefore it is crucial to analyzethe seepage pressure of the fractured straight well in lowpermeability carbonate reservoirs Several studies have beendone on the partial (completing) degree of perforation forhomogeneous (heterogeneous) reservoirs In 1963 Warrenbuilt an expression of the seepage pressure for homogeneous

reservoirs [4] In 1973 Ramey and Gringarten used Greenrsquosfunction method to obtain the analysis solution of thepartially penetrating straight well fractures [5 6] In 1987Heucman and Abbaszaden analyzed the seepage pressuredynamic characteristics of the infinite large reservoirs [6] In1991 Raghavan andOzkan obtained the point source solutionto the partial seepage pressure [7] In 2000 Bui created thepressure point source solution for the double mediumhomogeneous reservoirs [8 9] Valko and Amini presenteda method of distributed volume sources to investigate ahorizontal well with multiple transverse fractures [10]Lin and Zhu developed a slab source method to evaluateperformance of horizontal wells with or without fractures[11] Chinese scholars also contributed a lot in this field FengWenguang and Ge Jiaqing conducted a detailed study on

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4570539 10 pageshttpsdoiorg10115520184570539

2 Mathematical Problems in Engineering

y

z

x

1 2 3 4

0

Figure 1 The schematic of multistage fracturing vertical wells

single medium and double medium of non-Darcy flow anddraw a map of dynamic pressure characteristic curve [12]Fanhua and Ciqun obtained the unsteady seepage pressureexpression with considering start-up pressure gradient [13]Liu Qiguon and Wang Jianping used the boundary elementmethod to analyze the problem of unstable reservoir seepagein complex reservoirs [14 15] Zhao Y obtained an analyticalsolution of the seepage pressure in Lattice space domainwith taking reservoir heterogeneity into consideration[16 17]

However hydraulic fractures are assumed to be fullypenetrating the formation in the previous studies Limitedefforts have been made to investigate the effects of partiallypenetrating fracture height on the performance of wells Inpractice fully penetrating fractures may lead to an earlyor immediate water or gas breakthrough in a reservoirwith bottom water or gas cap in contact whereas partiallypenetrating fractures provide a better way to prevent theearly breakthrough What is more most of the previousmethods are based on Gringarten and Rameyrsquos point sourcesolution and Green function whereas the original physicalmodel established by Gringarten and Ramey only consideredthe upper and lower bounds limiting the scope of theapplication Therefore it is necessary to study flow pres-sure for low permeability oil well with partially penetratingfracture

This paper first presents the physical model and themathematical model of the unstable seepage flow in thethree-dimensional anisotropic rectangular reservoir undercertain conditions and then the solution of the mathematicalmodel is obtained which provides a new way to calculateflow pressure for low permeability oil well with partiallypenetrating fracture

2 Model Establishing

The reservoir formation is fractured by hydraulic fractur-ing to form a plurality of fractures as shown in Figure 1After hydraulic fracture the half-length and height of thefracture constitute the fracture surface Combining withthe distance between the fractures the one-dimensional

z

x

y

y

z

x

(cＲ cＳ cＴ)

Figure 2 Single fracture schematic of section penetration

diffusion volume of the reservoir fluid is built The dif-fusion volume affects the flow rate of the reservoir fluidin the longitudinal direction The fracture width and thefracture surface form the range of the linear flow in thefractures which affects the whole straight well productionThe influence of the fracture parameters on the seepagepressure can be analyzed according to the coupling modelof the reservoir-fracture-wellbore and through sensitivityanalyzing artificial fracture parameter combination can beobtained which forms the optimal fracture grid to achievethe purpose of yield optimizationTherefore in order to studythe seepage pressure of fractured straight well the seepagepressure of single fracture has to be studied as shown inFigure 2 and then by using the superposition principle themultifracture seepage pressure can be obtained In order tosimplify the physical model the assumptions are made asfollows

(1) The oil production is a constant and the formation ofthe reservoir is bounded and nonhomogeneous withequal thickness

(2) Before producing the reservoir pressure is equal tothe original formation pressure

(3) The reservoir fluid is microcompressible occurringsingle-phase and unstable seepage

(4) The intercrack interference is ignored and the fluidflow in the fracture obeys the Darcy law

(5) The gravity of the fluid and capillary force is ignoredand the porosity and fluid viscosity is constant

(6) The fracture is partially penetrating the formationand the reservoir fluid flows to the wellbore in alimited range

(7) Crossflow between the fracture and the matrix isignored and the fracture has infinite diversion capa-bility

The center of one single fracture is located at (1199090 1199100 1199110)based on the above assumptions the mathematical model ofthe fracture is established as follows

Mathematical Problems in Engineering 3

11989611990912060112058311988811990512059721198751205971199092 +

11989611991012060112058311988811990512059721198751205971199102 + 119896119911120601120583119888119905

12059721198751205971199112+ 8119902119908119909119908119910119908119911120601120583119888119905 119891 (119909 119910 119911) = 120597119875120597119905

(1)

where 119896119909 119896119910 and 119896119911 are in the permeability in the 119909 119910 and 119911direction 119888119905 is total compressibility120601 is porosity119901 is reservoirpressure 119905 is the time 119861 is formation volume factor and 119902 isthe flow rate of per unit area flowing through the fracture119891(119909 119910 119911) is the position of the source (sink) phase andits expression is written as follows119891 (119909 119910 119911)= 18119908119909119908119910119908119911∭V

120575 [(119909 minus 1199090) (119910 minus 1199100) (119911 minus 1199110)] 119889V(1199090 minus 119908119909) (1199100 minus 119908119910) (1199110 minus 119908119911) le V

le (1199090 + 119908119909) (1199100 + 119908119910) (1199110 + 119908119911)

(2)

where 120575 is Dirac Delta functionThe initial conditions are written as follows

119901 (119909 119910 119911 0) = 119901119894 (3)The inner boundary condition is written as follows

lim119903rarr119903119908

(119903120597119901120597119903) = 1198611199021205832120587119896ℎ119903 = radic(119909 minus 1199090)2 + (119909 minus 1199090)2 + (119909 minus 1199090)2

(4)

The outer boundary condition is written as follows12059711990112059711990910038161003816100381610038161003816100381610038161003816119909=0119886 = 0

12059711990112059711991010038161003816100381610038161003816100381610038161003816119910=0119887 = 0

12059711990112059711991110038161003816100381610038161003816100381610038161003816119911=0ℎ = 0

(5)

3 Model Solving

31 Solution Research According to the literature surveythere are mainly three methods to solve equation (1) Firstby ignoring the pressure propagation time and using Greenrsquosfunction method the analytical solution of the Lagrangianspace to (1) can be obtained [10 11 18] Secondly on thecondition that the boundary extension is regarded as thefunction of time by using steady state successive replacementmethod and differential discretizationmethod the numericalsolution to (1) can be obtained [13 16 19] Thirdly byusing numerical approximationmethod and series expansionmethod the correlation between the leading edge of thepressure propagation and the time can be solved A newway to solve (1) is present in this paper that is by meansof mathematical methods such as dimensionless transforma-tion Laplace transform Fourier cosine transformation andseparation variable method (1) can be solved

311 Dimensionless Transformation The dimensionlesstransformation is a method of converting the seepageequation into a conventional mathematical equation Bydimensionless transformation the number of comparisonscan be greatly reduced which makes the mathematicalphysics equation simple neat and easy to analyze and solveand help to check and verify the seepage equation [20] Thefollowing dimensionless transformation is introduced in thispaper

119873 = radic119886119887ℎ119871119875119863 = 2120587119896119873 (119901119894 minus 119901)119876119861120583119905119863 = 1198961206011205831198881199051198732 119905

119909119863 = radic 119896119896119909119909119873

119910119863 = radic 119896119896119909119910119873

119911119863 = radic 119896119896119909119911119873

119886119863 = 119886119873119887119863 = 119887119873ℎ119863 = ℎ119873119908119909119863 = 119908119909119873119908119910119863 = 119908119910119873119908119911119863 = 1199081199111198731199090119863 = 11990901199031199100119863 = 11991001199031199110119863 = 1199110119903

(6)

where 119871 is the length of the firing reservoir [m]So (1) can be written as

12059721198751198631205971199092119863 +12059721198751198631205971199102119863 +

12059721198751198631205971199112119863 +41205871198732119891 (119909119863 119910119863 119911119863) = 120597119875119863120597119905119863 (7)

4 Mathematical Problems in Engineering

where 119891(119909119863 119910119863 119911119863) is119891 (119909119863 119910119863 119911119863) = 18119908119909119863119908119910119863119908119911119863sdot∭

V119863120575 [(119909119863 minus 1199090119863) (119910119863 minus 1199100119863) (119911119863 minus 1199110119863)] 119889V119863

(1199090119863 minus 119908119909119863) (1199100119863 minus 119908119910119863) (1199110119863 minus 119908119911119863) le V119863

le (1199090119863 + 119908119909119863) (1199100119863 + 119908119910119863) (1199110119863 + 119908119911119863)

(8)

Initial conditions are as follows

119901119863 (119909119863 119910119863 119911119863 0) = 0 (9)

Inner boundary conditions are as follows

lim119903119863rarr1

(119903119863120597119901119863120597119903119863) = minus1 119903119863 = 119903119873 (10)

Outer boundary conditions are as follows

12059711990111986312059711990911986310038161003816100381610038161003816100381610038161003816119909119863=0119886119863 = 0

12059711990111986312059711991011986310038161003816100381610038161003816100381610038161003816119910119863=0119887119863 = 0

12059711990111986312059711991111986310038161003816100381610038161003816100381610038161003816119911119863=0ℎ119863 = 0

(11)

312 Equation Solving Laplace transform can eliminate thepartial derivative of time from the unstable seepage equationand has been widely used to solve the problem of unstableseepage [20] By using Laplace transformation (7) can bewritten as

12059721199011198631205971199092119863 +12059721199011198631205971199102119863 +

12059721199011198631205971199112119863 +41205871199041198732119891 (119909119863 119910119863 119911119863) = 119904119901119863 (12)

By using Fourier cosine transformation of 119909119863 119910119863 and 119911119863(12) can be written as

119901119863 = 11199062119898 + V2119899 + 1199082119901 + 11990441205871199041198732119891 (119906119898 V119899 119908119901) (13)

where 119906119898 V119899 and 119908119901 are the solution of the followingequations

119906119898 tan 119906119898 minus 119886119863 = 0V119899 tan V119899 minus 119887119863 = 0119908119901 tan119908119901 minus ℎ119863 = 0119891 (119906119898 V119899 119908119901) = 1198651 times 1198652 times 1198653

1198651 = sin [119906119898 (1199090119863 + 119908119909119863)] minus sin [119906119898 (1199090119863 minus 119908119909119863)]21199061198981199081199091198631198652 = sin [V119899 (1199100119863 + 119908119910119863)] minus sin [V119899 (1199100119863 minus 119908119910119863)]2V1198991199081199101198631198653 = sin [119908119901 (1199110119863 + 119908119911119863)] minus sin [119908119901 (1199110119863 minus 119908119911119863)]2119908119901119908119911119863

(14)

The Laplace space solution is obtained by using Fouriercosine inverse transformation The Fourier cosine inversetransformation can be written as

119875119863 (119909119863) = sum119898=1

cos (119906119898119909119863)119873 (119899) 119875119863 (119906119898) (15)

where119873(119899) is the norm and its expression is

119873(119899) = 12 (1 + sin 119906119899 cos 119906119899119906119899 ) (16)

By using Fourier cosine inverse transformation of 119911119863119910119863 and119909119863 (13) can be written as

119904119901119863 = 4120587119886119863119887119863ℎ1198631119904 + 2sum119898=1cos (119906119898119909119863)

1198651(119904 + 1199062119898)+ 2sum119899=1

cos (V119899119910119863) 1198652(119904 + V2119899)+ 4sum119899=1

sum119898=1

cos (V119899119910119863) cos (119906119898119909119863) 11986511198652(119904 + 1199062119898 + V2119899)+ 2sum119901=1

cos (119908119901119911119863) 1198653(119904 + 1199082119901)+ 4sum119898=1

sum119901=1

cos (119908119901119911119863) cos (119906119898119909119863) 11986511198653(119904 + 1199062119898 + 1199082119901)+ 4sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) 11986521198653(119904 + V2119899 + 1199082119901)+ 8sum119898=1

sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) cos (119906119898119909119863)

sdot 119865111986521198653(119904 + 1199062119898 + V2119899 + 1199082119901)

(17)

By introducing two equations

2 cos (120572120573) cos (120572120574) = cos [120572 (120573 + 120574)]+ cos [120572 (120573 minus 120574)]

sum119896=1

cos 1198961199091198962 + 1205722 = 1205872120572 cosh120572 (120587 minus 119909)sinh120572120587 minus 121205722 (18)

Mathematical Problems in Engineering 5

(17) can be written as

119904119901119863 = 2120587119886119863119887119863ℎ119863 [coshradic119904 (119887119863 minus 119910119863 + 1199100119863) + coshradic119904 (119887119863 minus 119910119863 minus 1199100119863)]radic119904 sinhradic119904119887119863 + 2sum

119898=1

cos (119906119898119909119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863times cosh 120591119898 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898 [119887119863 minus (119910119863 + 1199100119863)]120591119898 sinh 120591119898119887119863 + 2sum

119901=1

cos (119908119901119911119863) cos (1199081199011199100119863) sin (119908119901119908119911119863)119908119901119908119911119863times cosh 120591119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119901 [119887119863 minus (119910119863 + 1199100119863)]120591119901 sinh 120591119901119887119863+ 4 infinsum119898=1

infinsum119901=1

cos (119906119898119909119863) cos (119908119901119911119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863cos (1199081199011199110119863) sin (119908119901119908119911119863)119908119901119908119911119863

times cosh 120591119898119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898119901 [119887119863 minus (119910119863 + 1199100119863)]120591119898119901 sinh 120591119898119901119887119863

(19)

where

120591119898 = radic119904 + 1199062119898120591119901 = radic119904 + 1199082119901120591119898119901 = radic119904 + 1199062119898 + 1199082119901

(20)

Equation (19) is an analytical solutionmodel under Laplacianspace Its applicable conditions 119904 are an anisotropic homo-geneous rectangular reservoir with impermeable region andouter boundary closed by fracturing By using Stehfestnumerical inversion method [21] a numerical solution of theseepage pressure can be obtained By changing the center ofthe fracture the seepage pressure of the fracture at differentlocations can be obtained

32 Model Validation The mathematical model establishedin this paper can solve the pressure of multifracture systemFor the sake of simplification it is assumed that the numberof fracture is 10The numerical simulation is used to calculatethe seepage field and then the pressure value (simulatedsolution) at different time and different positions is outputand compared with the numerical solution of the seepagepressure calculated by Stehfest numerical inversion (thiswork)

The reservoir E300 module in Eclipse 2011 is developedfor fractured heterogeneous reservoirs E300 is used tosimulate the pressure variation around a fractured verticalwell in a rectangular heterogeneous reservoir In order tomeet the assumptions of (1) the numerical model is set asfollows

The width and the length of the rectangular heteroge-neous reservoir are 1 km and there is an oil productionwell inthe center of the reservoir which is showed as in Figure 3(a)A five-point well pattern is used to simulate the production

that is one production well in the reservoir center and fourinjectors in the four corners to ensure the production ofthe production well By adjusting injection volume the oilwell production under different displacement pressure can beobtained

In order to describe the formation fluid heterogeneity thetriangular network of grid system is used to ensure that eachcrack at least has 3 grids which is showed in Figure 3(b)Therefore the plane is divided into 20 lowast 20 meshes and theaverage grid step is 50 meters As (1) describing a single-phase fluid seepage only 1 simulation layer is divided in thevertical direction of the reservoir according to the seepagecharacteristics of single-phase fluid seepage The simulationof the total number of grid 20 lowast 20 lowast 1 = 4000 The requiredparameters for the numerical simulation are shown inTable 1and the results of two methods are shown in Table 2 FromTable 2 it can be seen that the relative error is under the basiccontrol of 5 which is in consistent with the allowable errorrange suggesting that this method we offered is reliable

33 Flow Period Division The fracture center is located onthe centerline of the wellbore axis and the partially pene-trating degree is 50 Moreover 119896119909 = 119896119910 and 119896119909119896119911 = 100There is a fracture in the center of the reservoir In order tostudy the trend of the bottom seepage pressure and pressurederivative the flow division schematic of partial penetrationfractured vertical wells is drawn as shown in Figure 4 Thepartially penetrating fracturing straight well seepage pressure(pressure derivative) model curve can be divided into fourflow periods A (early linear flow) B (medium radial flow) C(late spherical flow) andD (boundary control flow)The earlylinear flow is affected by epidermal effect andwellbore storageeffect and progressive analysis shows that during stage Athe slope of the pressure derivative curve is about 1 andthe reservoir fluid is continuously infiltrated to the wellboreand then the reservoir fluid enters the medium-term radialflow The higher the degree of partially penetrating is the

6 Mathematical Problems in Engineering

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(a)

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(b)

Figure 3 The geometry information representation of the reservoir

Table 1 Basic data of the system

parameters valuesaturation pressure 25MPaoil viscosity 121mPasdotsoil density 079 gcm3

water compressibility 49 times 10minus4MPaminus1

oil volume coefficient 121m3m3

porosity 012injection pressure 46MPaeffective thickness 5mformation temperature 158∘Fwater viscosity 16mPasdotsdissolved gas and oil ratio 2231m3m3

oil compressibility 81 times 10minus4MPaminus1

rock Compressibility 45 times 10minus4MPaminus1

permeability 12mDoriginal formation pressure 27MPainjecting water intensity 0044m3(dsdotMPasdotm)

longer themedium radial flow isWith the pressure graduallyspreading out before the pressure transmits to the boundaryit is mainly the late spherical flow and the pressure derivativegradient becomes smaller until it tends to be stableWhen thepressure propagates to the boundary the boundary controlflow occurs and the pressure derivative value changes fasterFinally the reservoir fluid flows to the wellbore in the form ofquasi-steady flow

4 Sensitivity Analysis

Based on the control variable method the parameters affect-ing pressure and pressure derivative (template curve) such as

A B C

D

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 4 The flow division schematic of partial penetrationfractured vertical wells

fracture orientation fracture scale the degree of penetrationin the reservoir permeability anisotropy reservoir boundarycondition and reservoir scale were analyzed by using theparameters of Table 1

41 Fracture Orientation In this paper the orientation offractures is divided into two aspects namely in the reservoircenter (as shown in Figure 5(a)) and not in the reservoircenter (Figure 5(b)) By setting the coordinates of differentfractures the relationship between the reciprocal pressurethe reciprocal of pressure and the producing time is shownas Figure 6

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

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Mathematical Problems in Engineering 3

11989611990912060112058311988811990512059721198751205971199092 +

11989611991012060112058311988811990512059721198751205971199102 + 119896119911120601120583119888119905

12059721198751205971199112+ 8119902119908119909119908119910119908119911120601120583119888119905 119891 (119909 119910 119911) = 120597119875120597119905

(1)

where 119896119909 119896119910 and 119896119911 are in the permeability in the 119909 119910 and 119911direction 119888119905 is total compressibility120601 is porosity119901 is reservoirpressure 119905 is the time 119861 is formation volume factor and 119902 isthe flow rate of per unit area flowing through the fracture119891(119909 119910 119911) is the position of the source (sink) phase andits expression is written as follows119891 (119909 119910 119911)= 18119908119909119908119910119908119911∭V

120575 [(119909 minus 1199090) (119910 minus 1199100) (119911 minus 1199110)] 119889V(1199090 minus 119908119909) (1199100 minus 119908119910) (1199110 minus 119908119911) le V

le (1199090 + 119908119909) (1199100 + 119908119910) (1199110 + 119908119911)

(2)

where 120575 is Dirac Delta functionThe initial conditions are written as follows

119901 (119909 119910 119911 0) = 119901119894 (3)The inner boundary condition is written as follows

lim119903rarr119903119908

(119903120597119901120597119903) = 1198611199021205832120587119896ℎ119903 = radic(119909 minus 1199090)2 + (119909 minus 1199090)2 + (119909 minus 1199090)2

(4)

The outer boundary condition is written as follows12059711990112059711990910038161003816100381610038161003816100381610038161003816119909=0119886 = 0

12059711990112059711991010038161003816100381610038161003816100381610038161003816119910=0119887 = 0

12059711990112059711991110038161003816100381610038161003816100381610038161003816119911=0ℎ = 0

(5)

3 Model Solving

31 Solution Research According to the literature surveythere are mainly three methods to solve equation (1) Firstby ignoring the pressure propagation time and using Greenrsquosfunction method the analytical solution of the Lagrangianspace to (1) can be obtained [10 11 18] Secondly on thecondition that the boundary extension is regarded as thefunction of time by using steady state successive replacementmethod and differential discretizationmethod the numericalsolution to (1) can be obtained [13 16 19] Thirdly byusing numerical approximationmethod and series expansionmethod the correlation between the leading edge of thepressure propagation and the time can be solved A newway to solve (1) is present in this paper that is by meansof mathematical methods such as dimensionless transforma-tion Laplace transform Fourier cosine transformation andseparation variable method (1) can be solved

311 Dimensionless Transformation The dimensionlesstransformation is a method of converting the seepageequation into a conventional mathematical equation Bydimensionless transformation the number of comparisonscan be greatly reduced which makes the mathematicalphysics equation simple neat and easy to analyze and solveand help to check and verify the seepage equation [20] Thefollowing dimensionless transformation is introduced in thispaper

119873 = radic119886119887ℎ119871119875119863 = 2120587119896119873 (119901119894 minus 119901)119876119861120583119905119863 = 1198961206011205831198881199051198732 119905

119909119863 = radic 119896119896119909119909119873

119910119863 = radic 119896119896119909119910119873

119911119863 = radic 119896119896119909119911119873

119886119863 = 119886119873119887119863 = 119887119873ℎ119863 = ℎ119873119908119909119863 = 119908119909119873119908119910119863 = 119908119910119873119908119911119863 = 1199081199111198731199090119863 = 11990901199031199100119863 = 11991001199031199110119863 = 1199110119903

(6)

where 119871 is the length of the firing reservoir [m]So (1) can be written as

12059721198751198631205971199092119863 +12059721198751198631205971199102119863 +

12059721198751198631205971199112119863 +41205871198732119891 (119909119863 119910119863 119911119863) = 120597119875119863120597119905119863 (7)

4 Mathematical Problems in Engineering

where 119891(119909119863 119910119863 119911119863) is119891 (119909119863 119910119863 119911119863) = 18119908119909119863119908119910119863119908119911119863sdot∭

V119863120575 [(119909119863 minus 1199090119863) (119910119863 minus 1199100119863) (119911119863 minus 1199110119863)] 119889V119863

(1199090119863 minus 119908119909119863) (1199100119863 minus 119908119910119863) (1199110119863 minus 119908119911119863) le V119863

le (1199090119863 + 119908119909119863) (1199100119863 + 119908119910119863) (1199110119863 + 119908119911119863)

(8)

Initial conditions are as follows

119901119863 (119909119863 119910119863 119911119863 0) = 0 (9)

Inner boundary conditions are as follows

lim119903119863rarr1

(119903119863120597119901119863120597119903119863) = minus1 119903119863 = 119903119873 (10)

Outer boundary conditions are as follows

12059711990111986312059711990911986310038161003816100381610038161003816100381610038161003816119909119863=0119886119863 = 0

12059711990111986312059711991011986310038161003816100381610038161003816100381610038161003816119910119863=0119887119863 = 0

12059711990111986312059711991111986310038161003816100381610038161003816100381610038161003816119911119863=0ℎ119863 = 0

(11)

312 Equation Solving Laplace transform can eliminate thepartial derivative of time from the unstable seepage equationand has been widely used to solve the problem of unstableseepage [20] By using Laplace transformation (7) can bewritten as

12059721199011198631205971199092119863 +12059721199011198631205971199102119863 +

12059721199011198631205971199112119863 +41205871199041198732119891 (119909119863 119910119863 119911119863) = 119904119901119863 (12)

By using Fourier cosine transformation of 119909119863 119910119863 and 119911119863(12) can be written as

119901119863 = 11199062119898 + V2119899 + 1199082119901 + 11990441205871199041198732119891 (119906119898 V119899 119908119901) (13)

where 119906119898 V119899 and 119908119901 are the solution of the followingequations

119906119898 tan 119906119898 minus 119886119863 = 0V119899 tan V119899 minus 119887119863 = 0119908119901 tan119908119901 minus ℎ119863 = 0119891 (119906119898 V119899 119908119901) = 1198651 times 1198652 times 1198653

1198651 = sin [119906119898 (1199090119863 + 119908119909119863)] minus sin [119906119898 (1199090119863 minus 119908119909119863)]21199061198981199081199091198631198652 = sin [V119899 (1199100119863 + 119908119910119863)] minus sin [V119899 (1199100119863 minus 119908119910119863)]2V1198991199081199101198631198653 = sin [119908119901 (1199110119863 + 119908119911119863)] minus sin [119908119901 (1199110119863 minus 119908119911119863)]2119908119901119908119911119863

(14)

The Laplace space solution is obtained by using Fouriercosine inverse transformation The Fourier cosine inversetransformation can be written as

119875119863 (119909119863) = sum119898=1

cos (119906119898119909119863)119873 (119899) 119875119863 (119906119898) (15)

where119873(119899) is the norm and its expression is

119873(119899) = 12 (1 + sin 119906119899 cos 119906119899119906119899 ) (16)

By using Fourier cosine inverse transformation of 119911119863119910119863 and119909119863 (13) can be written as

119904119901119863 = 4120587119886119863119887119863ℎ1198631119904 + 2sum119898=1cos (119906119898119909119863)

1198651(119904 + 1199062119898)+ 2sum119899=1

cos (V119899119910119863) 1198652(119904 + V2119899)+ 4sum119899=1

sum119898=1

cos (V119899119910119863) cos (119906119898119909119863) 11986511198652(119904 + 1199062119898 + V2119899)+ 2sum119901=1

cos (119908119901119911119863) 1198653(119904 + 1199082119901)+ 4sum119898=1

sum119901=1

cos (119908119901119911119863) cos (119906119898119909119863) 11986511198653(119904 + 1199062119898 + 1199082119901)+ 4sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) 11986521198653(119904 + V2119899 + 1199082119901)+ 8sum119898=1

sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) cos (119906119898119909119863)

sdot 119865111986521198653(119904 + 1199062119898 + V2119899 + 1199082119901)

(17)

By introducing two equations

2 cos (120572120573) cos (120572120574) = cos [120572 (120573 + 120574)]+ cos [120572 (120573 minus 120574)]

sum119896=1

cos 1198961199091198962 + 1205722 = 1205872120572 cosh120572 (120587 minus 119909)sinh120572120587 minus 121205722 (18)

Mathematical Problems in Engineering 5

(17) can be written as

119904119901119863 = 2120587119886119863119887119863ℎ119863 [coshradic119904 (119887119863 minus 119910119863 + 1199100119863) + coshradic119904 (119887119863 minus 119910119863 minus 1199100119863)]radic119904 sinhradic119904119887119863 + 2sum

119898=1

cos (119906119898119909119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863times cosh 120591119898 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898 [119887119863 minus (119910119863 + 1199100119863)]120591119898 sinh 120591119898119887119863 + 2sum

119901=1

cos (119908119901119911119863) cos (1199081199011199100119863) sin (119908119901119908119911119863)119908119901119908119911119863times cosh 120591119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119901 [119887119863 minus (119910119863 + 1199100119863)]120591119901 sinh 120591119901119887119863+ 4 infinsum119898=1

infinsum119901=1

cos (119906119898119909119863) cos (119908119901119911119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863cos (1199081199011199110119863) sin (119908119901119908119911119863)119908119901119908119911119863

times cosh 120591119898119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898119901 [119887119863 minus (119910119863 + 1199100119863)]120591119898119901 sinh 120591119898119901119887119863

(19)

where

120591119898 = radic119904 + 1199062119898120591119901 = radic119904 + 1199082119901120591119898119901 = radic119904 + 1199062119898 + 1199082119901

(20)

Equation (19) is an analytical solutionmodel under Laplacianspace Its applicable conditions 119904 are an anisotropic homo-geneous rectangular reservoir with impermeable region andouter boundary closed by fracturing By using Stehfestnumerical inversion method [21] a numerical solution of theseepage pressure can be obtained By changing the center ofthe fracture the seepage pressure of the fracture at differentlocations can be obtained

32 Model Validation The mathematical model establishedin this paper can solve the pressure of multifracture systemFor the sake of simplification it is assumed that the numberof fracture is 10The numerical simulation is used to calculatethe seepage field and then the pressure value (simulatedsolution) at different time and different positions is outputand compared with the numerical solution of the seepagepressure calculated by Stehfest numerical inversion (thiswork)

The reservoir E300 module in Eclipse 2011 is developedfor fractured heterogeneous reservoirs E300 is used tosimulate the pressure variation around a fractured verticalwell in a rectangular heterogeneous reservoir In order tomeet the assumptions of (1) the numerical model is set asfollows

The width and the length of the rectangular heteroge-neous reservoir are 1 km and there is an oil productionwell inthe center of the reservoir which is showed as in Figure 3(a)A five-point well pattern is used to simulate the production

that is one production well in the reservoir center and fourinjectors in the four corners to ensure the production ofthe production well By adjusting injection volume the oilwell production under different displacement pressure can beobtained

In order to describe the formation fluid heterogeneity thetriangular network of grid system is used to ensure that eachcrack at least has 3 grids which is showed in Figure 3(b)Therefore the plane is divided into 20 lowast 20 meshes and theaverage grid step is 50 meters As (1) describing a single-phase fluid seepage only 1 simulation layer is divided in thevertical direction of the reservoir according to the seepagecharacteristics of single-phase fluid seepage The simulationof the total number of grid 20 lowast 20 lowast 1 = 4000 The requiredparameters for the numerical simulation are shown inTable 1and the results of two methods are shown in Table 2 FromTable 2 it can be seen that the relative error is under the basiccontrol of 5 which is in consistent with the allowable errorrange suggesting that this method we offered is reliable

33 Flow Period Division The fracture center is located onthe centerline of the wellbore axis and the partially pene-trating degree is 50 Moreover 119896119909 = 119896119910 and 119896119909119896119911 = 100There is a fracture in the center of the reservoir In order tostudy the trend of the bottom seepage pressure and pressurederivative the flow division schematic of partial penetrationfractured vertical wells is drawn as shown in Figure 4 Thepartially penetrating fracturing straight well seepage pressure(pressure derivative) model curve can be divided into fourflow periods A (early linear flow) B (medium radial flow) C(late spherical flow) andD (boundary control flow)The earlylinear flow is affected by epidermal effect andwellbore storageeffect and progressive analysis shows that during stage Athe slope of the pressure derivative curve is about 1 andthe reservoir fluid is continuously infiltrated to the wellboreand then the reservoir fluid enters the medium-term radialflow The higher the degree of partially penetrating is the

6 Mathematical Problems in Engineering

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(a)

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(b)

Figure 3 The geometry information representation of the reservoir

Table 1 Basic data of the system

parameters valuesaturation pressure 25MPaoil viscosity 121mPasdotsoil density 079 gcm3

water compressibility 49 times 10minus4MPaminus1

oil volume coefficient 121m3m3

porosity 012injection pressure 46MPaeffective thickness 5mformation temperature 158∘Fwater viscosity 16mPasdotsdissolved gas and oil ratio 2231m3m3

oil compressibility 81 times 10minus4MPaminus1

rock Compressibility 45 times 10minus4MPaminus1

permeability 12mDoriginal formation pressure 27MPainjecting water intensity 0044m3(dsdotMPasdotm)

longer themedium radial flow isWith the pressure graduallyspreading out before the pressure transmits to the boundaryit is mainly the late spherical flow and the pressure derivativegradient becomes smaller until it tends to be stableWhen thepressure propagates to the boundary the boundary controlflow occurs and the pressure derivative value changes fasterFinally the reservoir fluid flows to the wellbore in the form ofquasi-steady flow

4 Sensitivity Analysis

Based on the control variable method the parameters affect-ing pressure and pressure derivative (template curve) such as

A B C

D

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 4 The flow division schematic of partial penetrationfractured vertical wells

fracture orientation fracture scale the degree of penetrationin the reservoir permeability anisotropy reservoir boundarycondition and reservoir scale were analyzed by using theparameters of Table 1

41 Fracture Orientation In this paper the orientation offractures is divided into two aspects namely in the reservoircenter (as shown in Figure 5(a)) and not in the reservoircenter (Figure 5(b)) By setting the coordinates of differentfractures the relationship between the reciprocal pressurethe reciprocal of pressure and the producing time is shownas Figure 6

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

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4 Mathematical Problems in Engineering

where 119891(119909119863 119910119863 119911119863) is119891 (119909119863 119910119863 119911119863) = 18119908119909119863119908119910119863119908119911119863sdot∭

V119863120575 [(119909119863 minus 1199090119863) (119910119863 minus 1199100119863) (119911119863 minus 1199110119863)] 119889V119863

(1199090119863 minus 119908119909119863) (1199100119863 minus 119908119910119863) (1199110119863 minus 119908119911119863) le V119863

le (1199090119863 + 119908119909119863) (1199100119863 + 119908119910119863) (1199110119863 + 119908119911119863)

(8)

Initial conditions are as follows

119901119863 (119909119863 119910119863 119911119863 0) = 0 (9)

Inner boundary conditions are as follows

lim119903119863rarr1

(119903119863120597119901119863120597119903119863) = minus1 119903119863 = 119903119873 (10)

Outer boundary conditions are as follows

12059711990111986312059711990911986310038161003816100381610038161003816100381610038161003816119909119863=0119886119863 = 0

12059711990111986312059711991011986310038161003816100381610038161003816100381610038161003816119910119863=0119887119863 = 0

12059711990111986312059711991111986310038161003816100381610038161003816100381610038161003816119911119863=0ℎ119863 = 0

(11)

312 Equation Solving Laplace transform can eliminate thepartial derivative of time from the unstable seepage equationand has been widely used to solve the problem of unstableseepage [20] By using Laplace transformation (7) can bewritten as

12059721199011198631205971199092119863 +12059721199011198631205971199102119863 +

12059721199011198631205971199112119863 +41205871199041198732119891 (119909119863 119910119863 119911119863) = 119904119901119863 (12)

By using Fourier cosine transformation of 119909119863 119910119863 and 119911119863(12) can be written as

119901119863 = 11199062119898 + V2119899 + 1199082119901 + 11990441205871199041198732119891 (119906119898 V119899 119908119901) (13)

where 119906119898 V119899 and 119908119901 are the solution of the followingequations

119906119898 tan 119906119898 minus 119886119863 = 0V119899 tan V119899 minus 119887119863 = 0119908119901 tan119908119901 minus ℎ119863 = 0119891 (119906119898 V119899 119908119901) = 1198651 times 1198652 times 1198653

1198651 = sin [119906119898 (1199090119863 + 119908119909119863)] minus sin [119906119898 (1199090119863 minus 119908119909119863)]21199061198981199081199091198631198652 = sin [V119899 (1199100119863 + 119908119910119863)] minus sin [V119899 (1199100119863 minus 119908119910119863)]2V1198991199081199101198631198653 = sin [119908119901 (1199110119863 + 119908119911119863)] minus sin [119908119901 (1199110119863 minus 119908119911119863)]2119908119901119908119911119863

(14)

The Laplace space solution is obtained by using Fouriercosine inverse transformation The Fourier cosine inversetransformation can be written as

119875119863 (119909119863) = sum119898=1

cos (119906119898119909119863)119873 (119899) 119875119863 (119906119898) (15)

where119873(119899) is the norm and its expression is

119873(119899) = 12 (1 + sin 119906119899 cos 119906119899119906119899 ) (16)

By using Fourier cosine inverse transformation of 119911119863119910119863 and119909119863 (13) can be written as

119904119901119863 = 4120587119886119863119887119863ℎ1198631119904 + 2sum119898=1cos (119906119898119909119863)

1198651(119904 + 1199062119898)+ 2sum119899=1

cos (V119899119910119863) 1198652(119904 + V2119899)+ 4sum119899=1

sum119898=1

cos (V119899119910119863) cos (119906119898119909119863) 11986511198652(119904 + 1199062119898 + V2119899)+ 2sum119901=1

cos (119908119901119911119863) 1198653(119904 + 1199082119901)+ 4sum119898=1

sum119901=1

cos (119908119901119911119863) cos (119906119898119909119863) 11986511198653(119904 + 1199062119898 + 1199082119901)+ 4sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) 11986521198653(119904 + V2119899 + 1199082119901)+ 8sum119898=1

sum119899=1

sum119901=1

cos (V119899119910119863) cos (119908119901119911119863) cos (119906119898119909119863)

sdot 119865111986521198653(119904 + 1199062119898 + V2119899 + 1199082119901)

(17)

By introducing two equations

2 cos (120572120573) cos (120572120574) = cos [120572 (120573 + 120574)]+ cos [120572 (120573 minus 120574)]

sum119896=1

cos 1198961199091198962 + 1205722 = 1205872120572 cosh120572 (120587 minus 119909)sinh120572120587 minus 121205722 (18)

Mathematical Problems in Engineering 5

(17) can be written as

119904119901119863 = 2120587119886119863119887119863ℎ119863 [coshradic119904 (119887119863 minus 119910119863 + 1199100119863) + coshradic119904 (119887119863 minus 119910119863 minus 1199100119863)]radic119904 sinhradic119904119887119863 + 2sum

119898=1

cos (119906119898119909119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863times cosh 120591119898 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898 [119887119863 minus (119910119863 + 1199100119863)]120591119898 sinh 120591119898119887119863 + 2sum

119901=1

cos (119908119901119911119863) cos (1199081199011199100119863) sin (119908119901119908119911119863)119908119901119908119911119863times cosh 120591119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119901 [119887119863 minus (119910119863 + 1199100119863)]120591119901 sinh 120591119901119887119863+ 4 infinsum119898=1

infinsum119901=1

cos (119906119898119909119863) cos (119908119901119911119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863cos (1199081199011199110119863) sin (119908119901119908119911119863)119908119901119908119911119863

times cosh 120591119898119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898119901 [119887119863 minus (119910119863 + 1199100119863)]120591119898119901 sinh 120591119898119901119887119863

(19)

where

120591119898 = radic119904 + 1199062119898120591119901 = radic119904 + 1199082119901120591119898119901 = radic119904 + 1199062119898 + 1199082119901

(20)

Equation (19) is an analytical solutionmodel under Laplacianspace Its applicable conditions 119904 are an anisotropic homo-geneous rectangular reservoir with impermeable region andouter boundary closed by fracturing By using Stehfestnumerical inversion method [21] a numerical solution of theseepage pressure can be obtained By changing the center ofthe fracture the seepage pressure of the fracture at differentlocations can be obtained

32 Model Validation The mathematical model establishedin this paper can solve the pressure of multifracture systemFor the sake of simplification it is assumed that the numberof fracture is 10The numerical simulation is used to calculatethe seepage field and then the pressure value (simulatedsolution) at different time and different positions is outputand compared with the numerical solution of the seepagepressure calculated by Stehfest numerical inversion (thiswork)

The reservoir E300 module in Eclipse 2011 is developedfor fractured heterogeneous reservoirs E300 is used tosimulate the pressure variation around a fractured verticalwell in a rectangular heterogeneous reservoir In order tomeet the assumptions of (1) the numerical model is set asfollows

The width and the length of the rectangular heteroge-neous reservoir are 1 km and there is an oil productionwell inthe center of the reservoir which is showed as in Figure 3(a)A five-point well pattern is used to simulate the production

that is one production well in the reservoir center and fourinjectors in the four corners to ensure the production ofthe production well By adjusting injection volume the oilwell production under different displacement pressure can beobtained

In order to describe the formation fluid heterogeneity thetriangular network of grid system is used to ensure that eachcrack at least has 3 grids which is showed in Figure 3(b)Therefore the plane is divided into 20 lowast 20 meshes and theaverage grid step is 50 meters As (1) describing a single-phase fluid seepage only 1 simulation layer is divided in thevertical direction of the reservoir according to the seepagecharacteristics of single-phase fluid seepage The simulationof the total number of grid 20 lowast 20 lowast 1 = 4000 The requiredparameters for the numerical simulation are shown inTable 1and the results of two methods are shown in Table 2 FromTable 2 it can be seen that the relative error is under the basiccontrol of 5 which is in consistent with the allowable errorrange suggesting that this method we offered is reliable

33 Flow Period Division The fracture center is located onthe centerline of the wellbore axis and the partially pene-trating degree is 50 Moreover 119896119909 = 119896119910 and 119896119909119896119911 = 100There is a fracture in the center of the reservoir In order tostudy the trend of the bottom seepage pressure and pressurederivative the flow division schematic of partial penetrationfractured vertical wells is drawn as shown in Figure 4 Thepartially penetrating fracturing straight well seepage pressure(pressure derivative) model curve can be divided into fourflow periods A (early linear flow) B (medium radial flow) C(late spherical flow) andD (boundary control flow)The earlylinear flow is affected by epidermal effect andwellbore storageeffect and progressive analysis shows that during stage Athe slope of the pressure derivative curve is about 1 andthe reservoir fluid is continuously infiltrated to the wellboreand then the reservoir fluid enters the medium-term radialflow The higher the degree of partially penetrating is the

6 Mathematical Problems in Engineering

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(a)

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(b)

Figure 3 The geometry information representation of the reservoir

Table 1 Basic data of the system

parameters valuesaturation pressure 25MPaoil viscosity 121mPasdotsoil density 079 gcm3

water compressibility 49 times 10minus4MPaminus1

oil volume coefficient 121m3m3

porosity 012injection pressure 46MPaeffective thickness 5mformation temperature 158∘Fwater viscosity 16mPasdotsdissolved gas and oil ratio 2231m3m3

oil compressibility 81 times 10minus4MPaminus1

rock Compressibility 45 times 10minus4MPaminus1

permeability 12mDoriginal formation pressure 27MPainjecting water intensity 0044m3(dsdotMPasdotm)

longer themedium radial flow isWith the pressure graduallyspreading out before the pressure transmits to the boundaryit is mainly the late spherical flow and the pressure derivativegradient becomes smaller until it tends to be stableWhen thepressure propagates to the boundary the boundary controlflow occurs and the pressure derivative value changes fasterFinally the reservoir fluid flows to the wellbore in the form ofquasi-steady flow

4 Sensitivity Analysis

Based on the control variable method the parameters affect-ing pressure and pressure derivative (template curve) such as

A B C

D

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 4 The flow division schematic of partial penetrationfractured vertical wells

fracture orientation fracture scale the degree of penetrationin the reservoir permeability anisotropy reservoir boundarycondition and reservoir scale were analyzed by using theparameters of Table 1

41 Fracture Orientation In this paper the orientation offractures is divided into two aspects namely in the reservoircenter (as shown in Figure 5(a)) and not in the reservoircenter (Figure 5(b)) By setting the coordinates of differentfractures the relationship between the reciprocal pressurethe reciprocal of pressure and the producing time is shownas Figure 6

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

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Mathematical Problems in Engineering 5

(17) can be written as

119904119901119863 = 2120587119886119863119887119863ℎ119863 [coshradic119904 (119887119863 minus 119910119863 + 1199100119863) + coshradic119904 (119887119863 minus 119910119863 minus 1199100119863)]radic119904 sinhradic119904119887119863 + 2sum

119898=1

cos (119906119898119909119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863times cosh 120591119898 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898 [119887119863 minus (119910119863 + 1199100119863)]120591119898 sinh 120591119898119887119863 + 2sum

119901=1

cos (119908119901119911119863) cos (1199081199011199100119863) sin (119908119901119908119911119863)119908119901119908119911119863times cosh 120591119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119901 [119887119863 minus (119910119863 + 1199100119863)]120591119901 sinh 120591119901119887119863+ 4 infinsum119898=1

infinsum119901=1

cos (119906119898119909119863) cos (119908119901119911119863) cos (1199061198981199090119863) sin (119906119898119908119909119863)119906119898119908119909119863cos (1199081199011199110119863) sin (119908119901119908119911119863)119908119901119908119911119863

times cosh 120591119898119901 [119887119863 minus (119910119863 minus 1199100119863)] + cosh 120591119898119901 [119887119863 minus (119910119863 + 1199100119863)]120591119898119901 sinh 120591119898119901119887119863

(19)

where

120591119898 = radic119904 + 1199062119898120591119901 = radic119904 + 1199082119901120591119898119901 = radic119904 + 1199062119898 + 1199082119901

(20)

Equation (19) is an analytical solutionmodel under Laplacianspace Its applicable conditions 119904 are an anisotropic homo-geneous rectangular reservoir with impermeable region andouter boundary closed by fracturing By using Stehfestnumerical inversion method [21] a numerical solution of theseepage pressure can be obtained By changing the center ofthe fracture the seepage pressure of the fracture at differentlocations can be obtained

32 Model Validation The mathematical model establishedin this paper can solve the pressure of multifracture systemFor the sake of simplification it is assumed that the numberof fracture is 10The numerical simulation is used to calculatethe seepage field and then the pressure value (simulatedsolution) at different time and different positions is outputand compared with the numerical solution of the seepagepressure calculated by Stehfest numerical inversion (thiswork)

The reservoir E300 module in Eclipse 2011 is developedfor fractured heterogeneous reservoirs E300 is used tosimulate the pressure variation around a fractured verticalwell in a rectangular heterogeneous reservoir In order tomeet the assumptions of (1) the numerical model is set asfollows

The width and the length of the rectangular heteroge-neous reservoir are 1 km and there is an oil productionwell inthe center of the reservoir which is showed as in Figure 3(a)A five-point well pattern is used to simulate the production

that is one production well in the reservoir center and fourinjectors in the four corners to ensure the production ofthe production well By adjusting injection volume the oilwell production under different displacement pressure can beobtained

In order to describe the formation fluid heterogeneity thetriangular network of grid system is used to ensure that eachcrack at least has 3 grids which is showed in Figure 3(b)Therefore the plane is divided into 20 lowast 20 meshes and theaverage grid step is 50 meters As (1) describing a single-phase fluid seepage only 1 simulation layer is divided in thevertical direction of the reservoir according to the seepagecharacteristics of single-phase fluid seepage The simulationof the total number of grid 20 lowast 20 lowast 1 = 4000 The requiredparameters for the numerical simulation are shown inTable 1and the results of two methods are shown in Table 2 FromTable 2 it can be seen that the relative error is under the basiccontrol of 5 which is in consistent with the allowable errorrange suggesting that this method we offered is reliable

33 Flow Period Division The fracture center is located onthe centerline of the wellbore axis and the partially pene-trating degree is 50 Moreover 119896119909 = 119896119910 and 119896119909119896119911 = 100There is a fracture in the center of the reservoir In order tostudy the trend of the bottom seepage pressure and pressurederivative the flow division schematic of partial penetrationfractured vertical wells is drawn as shown in Figure 4 Thepartially penetrating fracturing straight well seepage pressure(pressure derivative) model curve can be divided into fourflow periods A (early linear flow) B (medium radial flow) C(late spherical flow) andD (boundary control flow)The earlylinear flow is affected by epidermal effect andwellbore storageeffect and progressive analysis shows that during stage Athe slope of the pressure derivative curve is about 1 andthe reservoir fluid is continuously infiltrated to the wellboreand then the reservoir fluid enters the medium-term radialflow The higher the degree of partially penetrating is the

6 Mathematical Problems in Engineering

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(a)

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(b)

Figure 3 The geometry information representation of the reservoir

Table 1 Basic data of the system

parameters valuesaturation pressure 25MPaoil viscosity 121mPasdotsoil density 079 gcm3

water compressibility 49 times 10minus4MPaminus1

oil volume coefficient 121m3m3

porosity 012injection pressure 46MPaeffective thickness 5mformation temperature 158∘Fwater viscosity 16mPasdotsdissolved gas and oil ratio 2231m3m3

oil compressibility 81 times 10minus4MPaminus1

rock Compressibility 45 times 10minus4MPaminus1

permeability 12mDoriginal formation pressure 27MPainjecting water intensity 0044m3(dsdotMPasdotm)

longer themedium radial flow isWith the pressure graduallyspreading out before the pressure transmits to the boundaryit is mainly the late spherical flow and the pressure derivativegradient becomes smaller until it tends to be stableWhen thepressure propagates to the boundary the boundary controlflow occurs and the pressure derivative value changes fasterFinally the reservoir fluid flows to the wellbore in the form ofquasi-steady flow

4 Sensitivity Analysis

Based on the control variable method the parameters affect-ing pressure and pressure derivative (template curve) such as

A B C

D

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 4 The flow division schematic of partial penetrationfractured vertical wells

fracture orientation fracture scale the degree of penetrationin the reservoir permeability anisotropy reservoir boundarycondition and reservoir scale were analyzed by using theparameters of Table 1

41 Fracture Orientation In this paper the orientation offractures is divided into two aspects namely in the reservoircenter (as shown in Figure 5(a)) and not in the reservoircenter (Figure 5(b)) By setting the coordinates of differentfractures the relationship between the reciprocal pressurethe reciprocal of pressure and the producing time is shownas Figure 6

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(a)

0 200 400 600 800 1000

200

400

600

800

1000

y (m

)

x (m)

(b)

Figure 3 The geometry information representation of the reservoir

Table 1 Basic data of the system

parameters valuesaturation pressure 25MPaoil viscosity 121mPasdotsoil density 079 gcm3

water compressibility 49 times 10minus4MPaminus1

oil volume coefficient 121m3m3

porosity 012injection pressure 46MPaeffective thickness 5mformation temperature 158∘Fwater viscosity 16mPasdotsdissolved gas and oil ratio 2231m3m3

oil compressibility 81 times 10minus4MPaminus1

rock Compressibility 45 times 10minus4MPaminus1

permeability 12mDoriginal formation pressure 27MPainjecting water intensity 0044m3(dsdotMPasdotm)

longer themedium radial flow isWith the pressure graduallyspreading out before the pressure transmits to the boundaryit is mainly the late spherical flow and the pressure derivativegradient becomes smaller until it tends to be stableWhen thepressure propagates to the boundary the boundary controlflow occurs and the pressure derivative value changes fasterFinally the reservoir fluid flows to the wellbore in the form ofquasi-steady flow

4 Sensitivity Analysis

Based on the control variable method the parameters affect-ing pressure and pressure derivative (template curve) such as

A B C

D

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 4 The flow division schematic of partial penetrationfractured vertical wells

fracture orientation fracture scale the degree of penetrationin the reservoir permeability anisotropy reservoir boundarycondition and reservoir scale were analyzed by using theparameters of Table 1

41 Fracture Orientation In this paper the orientation offractures is divided into two aspects namely in the reservoircenter (as shown in Figure 5(a)) and not in the reservoircenter (Figure 5(b)) By setting the coordinates of differentfractures the relationship between the reciprocal pressurethe reciprocal of pressure and the producing time is shownas Figure 6

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

Table 2 The results of comparative table

production time(119905 d) Oil production(119902 m3d) Pressure of this work

(119901 MPa)Pressure of E300

(119901 MPa)relative error

()30 17492 22984 22249 320160 17202 22603 21870 324490 16951 22274 21539 3298120 16729 21982 21251 3323150 16532 21723 20993 3363180 16355 21490 20767 3369210 16194 21279 20558 3390240 16048 21087 20366 3421270 15913 20910 20196 3412300 15770 20723 19912 3913330 15627 20538 19731 3929360 15486 20353 19450 4436

x x

z z

x x

y y

(a)

x x

z z

x x

y y

(b)

Figure 5 Schematic diagram of fracture orientation

in the centernot in the center

in the centernot in the center

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 6 The effect of fracture orientation on template curves

Figure 6 shows that the fracture orientationmainly affectsthe late spherical flow stage of the template curve When thefracture is located in the reservoir center the pressure is easierto propagate outwards because in the late spherical flowstage the reservoir fluid flows to the wellbore in the form ofspace sinksThe fractures of hydraulic fracturing are the ldquoflow

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

fracture length 2= 300 mfracture length 2= 200 mfracture length 2= 100 m

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 7 The effect of fracture length on template curves

networkrdquo established in the reservoir and connect the seepagechannel of the reservoir which increases the seepage areaexposed to the reservoir The more symmetrical the fractureis to the center the more simple the fluid of the flow channelis and the easier the fluid flow and pressure transmission are

42 Fracture Length With different fracturing scale thelength of the fractures is not the sameWith different fracturelength the seepage area exposed to the reservoir is not thesame so the pressure transmission trend is different as shownin Figure 7 Figure 7 shows that the length of the fracturemainly affects the early linear and medium radial flowespecially the medium radial flow At the same productiontime as the length of the fracture increases the pressuredrop becomes slower and the pressure propagation becomesfaster and finally the pressure increases significantly In theearly linear flow stage small fractures tend to produce morepressure drop which is because under the condition of thesame production rate the larger the size of the fractures isthe bigger the seepage area is So the wellbore fluid supplycapacity is strong and the time of the early linear flow stage

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

penetration degree=100penetration degree=50penetration degree=30

penetration degree=100penetration degree=50penetration degree=30

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 8 The effect of opening shot degree on template curves

is longer When the fracture length is the same the pressuredrop velocity tends to be consistent so the fracture lengthonly affects the duration of the early linear flow and its effecton the pressure drop is not obvious Using the progressiveanalysis method we can see that the slope of the straight linein this stage is 05

43 The Degree of Penetration in the Reservoir Reasonabledegree of penetration in the reservoir not only can save thecost of perforation but also can get the maximum yield Theeffect of the degree of penetration in the reservoir on fluidpressure is obvious as shown in Figure 8 Figure 8 shows thatthe degree of penetration mainly affects the end of the latespherical flow and the beginning of the boundary controlflow When the height of the reservoir increases the timeof the late spherical flow becomes shorter and the fluidseepage enters the boundary control flow stage earlier Whenthe reservoir is completely penetrated the fluid flows intothe boundary control flow stage without going through thespherical flow stage

44 Reservoir Anisotropy Since the permeability varies littlein the horizontal direction the permeability in 119909 and 119910direction is considered as the same value and the effect ofthe vertical permeability anisotropy on the template curveis studied as shown in Figure 9 Figure 9 shows that thepermeability of the vertical anisotropy mainly affects themedium radial flow stage It is mainly because that in theactual formation the greater the vertical permeability is thelarger the probability of fluid flow between the fractures isOn the contrary it is smaller Therefore when the ratio ofhorizontal permeability to vertical permeability increases thetime of the medium radial flow is longer Because when thehorizontal permeability plays a dominant role the reservoirfluid mainly flows to the wellbore from the horizontaldirection

45 Reservoir Boundary Conditions The combination ofdifferent reservoir boundary conditions has different effect onthe template curve as shown in Figure 10 Figure 10 shows

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

ＥＢＥＰ=100ＥＢＥＰ=10ＥＢＥＰ=1

Figure 9 The effect of reservoir anisotropy on template curves

upper close lower closeupper close lower constantupper constant lower constant

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 10 The effect of reservoir boundary conditions on templatecurves

that the boundary condition mainly affects the boundarycontrol flow stage For the closed boundary the value of thepressure derivative tends to be stable and the progressiveanalysis shows that the pressure derivative is about 001 Forthe constant pressure boundary the pressure is continuouslypropagating and the pressure derivative is gradually reducingdue to sufficient supply of external energy until the pressureis equal to the boundary pressure The progressive analysisshows that the pressure derivative slope is about 1 and thereservoir fluid makes quasi-steady state seepage When theupper (lower) boundary is closed and the lower (upper)boundary is constant pressure then the pressure change isbetween the above two cases and only the pressure derivativeslope is not a constant

46 Reservoir Width By setting different reservoir widthsthe influence of reservoir width on the template curve isstudied as shown in Figure 11 Figure 11 shows that thereservoir width mainly affects the time when the fluid entersthe boundary control flow stage and the speed of the pressurederivative curve decreasing at this stage With the reservoirwidth decreasing the seepage pressure enters the boundary

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

reservoir width=1 kmreservoir width=3 kmreservoir width=5 km

ＦＨＪ＄

ln(＞Ｊ＄ＦＨＮ＄)

minus4

minus3

minus2

minus1

0

1

2

ＦＨＪ＄

ampln(＞

Ｊ＄＞

ＦＨＮ ＄

)

minus2 minus1minus3 1 2 3 4 50ＦＨＮ＄

Figure 11 The effect of reservoir width on template curves

control flow stage more quickly and the pressure drop issmaller The smaller the width of the reservoir is the morequickly the pressure spreads to the boundary under thecondition of constant production producing The seepagepressure must be into the boundary control flow stage earlierand finally the pressure increases earlier and faster and theproduction pressure increases faster tomaintain fluid flowingto the wellbore which is constant in the unit time

5 Conclusion

(1) The mathematical model of the unstable seepage flowin the three-dimensional anisotropic rectangular reservoiris deduced by establishing a physical model which is con-sistent with the actual formation of the nonhomogeneousreservoir The model considers the impermeable top bottomand constant pressure bottom boundary and other differ-ent boundary conditions combined with each other Thenumerical solution of the pressure in real domain of themodel is obtained by using Laplace transform Fourier cosinetransform and Stephenson numerical inversion method Thecalculation results are in good agreement with the numericalsimulation which proves the correctness of the model andthe practicability of the method(2)Thepressure dynamicmodel curve can be divided intofour flow periods early linear flow medium radial flow latespherical flow and boundary control flow Different reservoirphysical properties and different fracturing constructionscale are in varying degrees affecting the seepage pressureThe depth of the fracture mainly affects the early linear flowstageThe permeability anisotropymainly affects themediumradial flow stage The reservoir firing degree and the fractureorientation mainly affect the late spherical flow stage Theboundary conditions and the reservoir width mainly affectthe boundary control flow stage(3) In the early linear flow stage the pressure and pressurederivative curve are a straight line the scale of the fracturemainly affects the early linear flow and small fractures tendto produce larger pressure drop In the medium radial flowstage pressure and pressure derivative curve show radial flow

characteristics of finite extending wells in infinite extensionsystem and the pressure derivative is approximately parallelto the abscissa which is a constant The size of the imperme-able region and the orientation of the fracture determine thetime when the medium radial flow occurs and its durationIn the late spherical flow stage the degree of penetration andthe location of the fracture determine the time when the latespherical flow occurs and its duration The reservoir fluidflow is infinitely close to the quasi-steady state seepage flowand the rate of the pressure derivative is about 05 In theboundary control flow stage the pressure is affected by theboundary condition type and the reservoir width(4) This method can determine the parameters suchas optimal degree of penetration and vertical permeabilityand provide theoretical guidance for reservoir engineeringanalysis and fracturing process design

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S J Al Rbeawi and D Tiab ldquoEffect of penetrating ratio onpressure behavior of horizontal wells with multiple-inclinedhydraulic fracturesrdquo in Proceedings of the SPE Western RegionalMeeting Bakersfield Calif USA 2012

[2] S J Al Rbeawi and D Tiab ldquoPartially penetrating hydraulicfractures pressure responses and flowdynamicsrdquo inProceedingsof the SPE Production and Operations Symposium OklahomaCity Okla USA 2013

[3] O Alpheus and D Tiab ldquoPressure transient analysis in par-tially penetrating infinite conductivity hydraulic fractures innaturally fractured reservoirsrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colo USA 2008

[4] M Mills and M W Clegg ldquoStudy of behavior of partiallypenetrating wellsrdquo Society of Petroleum Engineers Journal (SPE)vol 2054 1969

[5] H Ramey and A C Gringarten ldquoThe use of source and Greenrsquosfunction in solving unsteady-flow problem in reservoirrdquo Societyof Petroleum Engineers Journal vol 13 no 5 1973

[6] A C Gringarten and H J Ramey ldquoUnsteady pressure dis-tribution created by a single horizontal fracture and partialpenetration or restricted entryrdquo Society of Petroleum EngineersJournal (SPE) vol 14 no 4 pp 413ndash426 1974

[7] M Buhidmal and R Raghavan ldquoTransient pressure of partiallypenetrating wells subject to bottom-water driverdquo Journal ofPetroleum Technology vol 32 no 7 1980

[8] F J Kuchuk and P A Kirwan ldquoNew skin and wellbore storagetype curves for partially penetrated wellsrdquo SPE FormationEvaluation vol 2 no 4 pp 546ndash554 1987

[9] M Abbaszadeh and P S Hegeman ldquoPressure-transient analysisfor a slanted well in a reservoir with vertical pressure supportrdquoSPE Formation Evaluation vol 5 no 3 pp 277ndash284 1990

[10] M Onur A Satman and A Reynolds ldquoNew type curves foranalyzing the transition time data from naturally fracturedreservoirsrdquo in Proceedings of the Low Permeability ReservoirsSymposium Denver Colo USA 1993

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

[11] T D Bui DDMamora andW J Lee ldquoTransient pressure anal-ysis for partially penetrating wells in naturally fractured reser-voirsrdquo in Proceedings of the SPE Rocky Mountain RegionalLowPermeability Reservoirs Symposium and Exbibition SPE PaperNo 60289 pp 1ndash8 Denver Colo USA March 2000

[12] F Wenguang and G Jiali ldquoThe non-Darcy flow problem ofunsteady state in a single media or dual mediardquo PetroleumExploration and Development vol 12 no 1 pp 56ndash62 1985

[13] L Fanhua and L Ciqun ldquoPressure transient analysis forunsteady porous flow with start-up pressure derivativerdquo WellTesting vol 6 no 1 pp 1ndash4 1997

[14] C Shiqing LGongquan L Tao et al ldquoMathematicalmodel andtypical curve for calculating effective hole diameter in the lowvelocity non-darcy flow testing of dual-media reservoirrdquo NGIvol 17 no 2 pp 35ndash37 1997

[15] S Fuquan and L Ciqun ldquoAnalasis of pressure and productionin the deformable porous mediardquo Petroleum Exploration andDevelopment vol 27 no 1 pp 57ndash59 2000

[16] L Qiguo L Xiaoping and W Xiaoqing ldquoAnalysis of pressuretransient behaviors in arbitrarily shaped reservoirs by theboundary element methodrdquo Journal of Southwest PetroleumInstitute vol 23 no 2 pp 40ndash43 2001

[17] L Qingshan D Yonggang W Chen et al ldquoApplication ofboundary element in unsteady state flowrdquo Petroleum Explo-ration and Development vol 23 no 2 pp 36-37 2004

[18] C Shiqing X Lunyun and Z Dechao ldquoType curve matchingof well test data for non-darcy flow at low velocityrdquo PetroleumExploration and Development vol 23 no 4 pp 50ndash53 1996

[19] W JianpingW Xiaodong andM Shidong ldquoUnstable filtrationtheory on straight well penetrating in various partsrdquo PetroleumExploration and Development vol 26 no 3 pp 65ndash71 2007

[20] W Xiaodong The Foundation of Seepage Mechanics ChinaUniversity of Geosciences Press Beijing China 2006

[21] H Stehfest ldquoNumerical inversion of Laplace transformsrdquo Com-munications of the ACM vol 20 no 1 pp 47-48 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom