Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 802585 6 pageshttpdxdoiorg1011552013802585

Research ArticleInverse Problem Models of Oil-Water Two-Phase Flow Based onBuckley-Leverett Theory

Rui Huang1 Xiaodong Wu1 Ruihe Wang2 and Hui Li3

1 College of Petroleum Engineering China University of Petroleum Beijing 102249 China2 China National Oil and Gas Exploration and Development Corporation Beijing 100034 China3 CNPC Beijing Richfit Information Technology Co Ltd Beijing 100013 China

Correspondence should be addressed to Rui Huang huangrui25126com

Received 5 September 2013 Revised 30 October 2013 Accepted 30 October 2013

Academic Editor Xiaoqiao He

Copyright copy 2013 Rui Huang 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

Based on Buckley-Leverett theory one inverse problemmodel of the oil-water relative permeability was modeled and proved whenthe oil-water relative permeability equations obey the exponential form expression and under the condition of the formationpermeability that natural logarithmdistribution always obey normal distribution the other inverse problemmodel on the formationpermeability was proved These inverse problem models have been assumed in up-scaling cases to achieve the equations byminimization of objective function different between calculation water cut and real water cut which can provide a reference forresearching oil-water two-phase flow theory and reservoir numerical simulation technology

1 Introduction

Reservoir numerical simulation technology is a growing newdiscipline with the emergence and development of the com-puter technology and computational mathematics which hasachieved rapid development and wide application all over theworld for example the study of reservoir numerical simu-lation based on formation parameters [1] the well modelsand impacts [2] a numerical simulator for low-permeabilityreservoirs [3] and so on However the majority of methodshave a great amount of calculation It will waste a lotof time and energy when we research the oil-water relativepermeability equations or the formation permeability distri-butionTherefore according to the inverse problemmodelingbased on oil-water two-phase flow we propose a method toobtain the relevant results for our research It is importantto define the formation permeability distribution and oil-water relative permeability equations which provide neces-sary information for oil reservoir evaluation for examplelarge-scaling evaluation which could be applied in reservoirnumerical simulation during evaluation

In order to get the answers which will be able to apply inlarge-scaling cases and solve the corresponding problems in

reservoir scaling inverse problemmodels have to be assumedto achieve the equations by minimization of objective func-tion differently between real water cut and calculation watercut The theoretical grid model is shown in Figure 1

If we know the distribution 1198701 1198702 119870

119899 then the

inverse problem model will be as shown in (1)The first question The optimization distribution

119886119908 119887119908 119886119900 119887119900 and the relative permeability equations

119870119903119908(119878119908)119870119903119900(119878119908)

The inverse problem mathematical model on oil-waterrelative permeability is as followsobjective function

119864 = min119899119905

sum119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2

where 119899119905 is time steps

initial condition

119878119908

119894(0)

119895= 119878119868119908119895

1198760

119895= 0

0 le 119870119903119900119894

119895lt 1 0 le 119870119903119908

119894

119895lt 1

2 Journal of Applied Mathematics

1234

n

234m

The average partition total m

The p

artit

ion

tota

l n

Supply edge Fluid producing edge

middot middot middot

re

Figure 1 Reservoir profile model

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

if 119903119890gt 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119878119868119908119895

if 119903119890le 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908)

119891119908= 119891119908(119878119908)

mathematical model

119870119903119900119894

119895= 119870119903119900 (119878

119908

119894(119905)

119895) 119870119903119908

119894

119895= 119870119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895 119876

119905

119895=119879119895

119879sdot 119876119905

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895 119891

119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(1)

If we know the equations 119870119903119908(119878119908) 119870119903119900(119878119908) another

inverse problem model will be obtained as shown in (2)The second question an optimization distribution prob-

lem 1198701 1198702 119870

119899 and the standard deviation 120590

The inverse problem mathematical model of the forma-tion permeability is as follows

objective function

119864 = min119899119905

sum119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2

where 119899119905 is time steps

initial condition

119878119908

119894(0)

119895= 119878119908119888119895

1198760

119895= 0

119891 (119870119895) =

1

radic2120587 sdot 120590119890minus(ln119870

119895minus120583)2

21205902

120583 = Ln119870

if 119903119890gt 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119878119908119888119895

if 119903119890le 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908)

119891119908= 119891119908(119878119908)

mathematical model

119870119903119900119894

119895= 119870119903119900 (119878

119908

119894(119905)

119895) 119870119903119908

119894

119895= 119870119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895 119876

119905

119895=119879119895

119879sdot 119876119905

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895 119891

119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(2)

Here 119891(history)119908

(119905) is history water cut and 119891119908(119905) is cal-

culation water cut the first water cut is a known amountfrom production and another is calculated by our inverseproblem models Water cut means water production rate is akey parameter in reservoir engineering which can representwater production capacity in reservoir development If watercut is too high it may bring negative effects to reservoirproduction Therefore the history matching for water pro-duction rate plays an important role in reservoir dynamicanalysis and numerical simulation [4ndash6]

And thus 119894 = 1 2 3 119898 119895 = 1 2 3 119899 119898

is the average partition total 119899 is the partition total 119879is grid conductivity ℎ

119895is each longitudinal layer stratum

Journal of Applied Mathematics 3

dr

Supp

ly ed

ge

Fluid producing edge

h

re

Figure 2 Radial flow unit model

thickness 119870119895is each longitudinal layer permeability 120601

119895is

each longitudinal layer porosity 119878119894119908119895

is grid water saturationat different time 119878

119908119888is the initial water saturation 119864 is

the objective function 119903119890is reservoir radius 119870 is average

permeability 119870119903119900

is oil relative permeability 119870119903119908

is waterrelative permeability119876

119905is the total fluid production119876119905

119908is the

total water yield of fluid producing edge at different time 119876119905119900

is the total oil production of fluid producing edge at differenttime 119876119905

119895is the total fluid production of each longitudinal

layer at different time 119878119908119891

is the frontier saturation 119903119908119891

is thecorresponding displacement of frontier saturation

The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032

119890minus 1199032= 1198911015840

119908sdot 119876119905120601 sdot 120587 sdot ℎ

without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus

119891(history)119908

(119905))2 By the end an inverse problem model on the

optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water

relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on

the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies

2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow

Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (3)

Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2

According to the seepage principle [9] we can get a flowequation of the volume element

119902119908= 119902 sdot 119891

119908 (4)

In the 119889119905 time the flow volume of the volume element is

119876out = 119889119902119908sdot 119889119905 (5)

We can derive from (4) and (5) that

119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)

Meanwhile the inflow volume of the volume element is

119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)

In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows

119902 sdot119889119891119908

119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)

From 119891119908= 119891119908(119878119908) we can get that 1198911015840

119908= 119889119891119908119889119878119908

Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903

after infinitesimal calculus we obtain

1198911015840

119908int119905

0

119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890

119903

119903119889119903 (9)

If119876119905 = int119905

0119902 sdot119889119905 then (9) turns to1198911015840

119908sdot119876119905= 120601sdot120587ℎ sdot (119903

2

119890minus1199032)

We obtain

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (10)

From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(11)

Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from

the water cut function curve 119891119908

= 119891119908(119878119908) selecting the

point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891

119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878

119908119891) then

one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding

displacement of frontier saturation 119903119908119891

is

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(12)

4 Journal of Applied Mathematics

3 The Inverse Problem Model on Oil-WaterRelative Permeability

31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(13)

In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed

119870119903119908

(119878119894

119908119895) = 119886119908(

119878119894

119908119895minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119894

119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894

119908119895

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(14)

In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-

sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)

32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations

Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the

water cut 119891119908(119905) based on the model of the Theorem 1 if the

objective function 119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 can

be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886

119908 119887119908 119886119900 119887119900 and oil-

water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)

Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following

initial value 1198860

119908 1198870

119908 1198860

119900 1198870

119900

119870119903119908

(119878119908) = 1198860

119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119908

119870119903119900(119878119908) = 1198860

119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119900

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(15)

We can derive from (12) and (15) 119903119908119891

If 119903119890gt 119903119908119891 then 119878

119908

119894(119905)

119895= 119878119908119888119895

If 119903119890le 119903119908119891

solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908) and so forth 119894 = 1

If we obtain the value of the water saturation [12] relyingon the function 119891

119908(119878119908) sim 119878

119908 we can calculate the value

of the longitudinal each layer water cut 119891119905119908(1198781

119908119895) and the

constructing mathematic model as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895)

119876119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(16)

From (16) we can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and rely on the objective function 119864 =

minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 with numerical optimization

calculation in the inverse problem model an optimizationproblem of the distribution 119886

119908 119887119908 119886119900 119887119900 will be obtained

so the 119870119903119908(119878119908) and 119870

119903119900(119878119908) can be formed The proof of

Theorem 3 is completed

4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution

Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870

1 Ln119870

2 Ln119870

119899 always obey nor-

mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =

Ln119870 meanwhile one calculates the value of the 119891119908(119905) based

on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905

119905=1(119891119908(119905)minus119891

(history)119908

(119905))2 then the

inverse problem model (2) will have the optimal distributionLn119870

1 Ln119870

2 Ln119870

119899 and 119870

1 1198702 119870

119899

Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)

According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min

According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883

0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a

cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for

Journal of Applied Mathematics 5

probability curve then the area summation of every closedfigures 119878

119895= 119865(119883

119895) minus119865(119883

119895minus1) each longitudinal layer stratum

thickness ℎ119895= ℎ sdot 119878

119895 and 119883

119895= Δ119883 sdot 119895 + 119883

0 119870119895= 119890119883119895

if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878

119895=

119865(119883119895)minus119865(119883

119895minus1) 119878119895= Δ119878 = 1119899 ℎ

119895= ℎsdot119878

119895 and119883

119895= 119865minus1(119878119895)

119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of

vertical stratification)Relying on the distribution 119870

1 1198702 119870

119899 oil-water

relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =

119896119903119908(119878119908) [14] the conductivity definition of numerical reser-

voir simulation we get the following equations

119896119903119900119894

119895= 119896119903119900 (119878

119908

119894(119905)

119895) 119896119903119908

119894

119895= 119896119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(17)

We can derive from (12) and (17) that 119903119908119891119895

If 119903119890

le 119903119908119891119895

solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908

119894(119905)

119895= 119865minus1(119903119894

119895 120601119895 ℎ119895 119876119905

119895) and thus 119894 = 1 If 119903

119890gt 119903119908119891119895

then

119878119908

119894(119905)

119895= 119878119908119888119895

and thus 119894 = 1If we obtain the value of the water saturation (119878

119908) relying

on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the

longitudinal each layer water cut 119891119905119908(1198781

119908119895) then mathematic

model can be constructed as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

(119876119905119908+ 119876119905119900)

(18)

Equation (18) can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and relies on the objective function

119864 = minsum119899119905119905=1

(119891119908(119905)minus119891

(history)119908

(119905))2With numerical optimiza-

tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870

119899 will be obtained so the distribution

1198701 1198702 119870

119899 will be given The proof of Theorem 4 is

completed

5 Discussions and Conclusions

The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a

group of oil-water relative permeability equations 119870119903119908(119878119908)

and 119870119903119900(119878119908) Combining with the objective function

119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 through the optimization

solution method it will finally bring a set of the optimumdistribution 119886

119908 119887119908 119886119900 119887119900 and the oil-water relative

permeability equations119870119903119908(119878119908) and119870

119903119900(119878119908)

According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583

119894 1205902

119894) corresponds to its own normal distribution

curve with the certain expectation 120583119894 If 120590

119894goes up the

volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583

119894

each different value of 120590119894corresponds to a group of original

values of (120583119894 1205902

119894) which will yield a group of values of

1198701 1198702 119870

119899minus1 119870119899 namely the value of permeability of

each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus 119891

(history)119908

(119905))2

it will finally bring a set of optimum normal distributionsof 119883 sim (120583

119894 1205902

119894) (Etc 119883 = ln119870) through the optimization

solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model

Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance

Acknowledgments

The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions

References

[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012

[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012

[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013

[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011

[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

1234

n

234m

The average partition total m

The p

artit

ion

tota

l n

Supply edge Fluid producing edge

middot middot middot

re

Figure 1 Reservoir profile model

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

if 119903119890gt 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119878119868119908119895

if 119903119890le 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908)

119891119908= 119891119908(119878119908)

mathematical model

119870119903119900119894

119895= 119870119903119900 (119878

119908

119894(119905)

119895) 119870119903119908

119894

119895= 119870119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895 119876

119905

119895=119879119895

119879sdot 119876119905

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895 119891

119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(1)

If we know the equations 119870119903119908(119878119908) 119870119903119900(119878119908) another

inverse problem model will be obtained as shown in (2)The second question an optimization distribution prob-

lem 1198701 1198702 119870

119899 and the standard deviation 120590

The inverse problem mathematical model of the forma-tion permeability is as follows

objective function

119864 = min119899119905

sum119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2

where 119899119905 is time steps

initial condition

119878119908

119894(0)

119895= 119878119908119888119895

1198760

119895= 0

119891 (119870119895) =

1

radic2120587 sdot 120590119890minus(ln119870

119895minus120583)2

21205902

120583 = Ln119870

if 119903119890gt 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119878119908119888119895

if 119903119890le 119903119908119891

119894

119895 then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908)

119891119908= 119891119908(119878119908)

mathematical model

119870119903119900119894

119895= 119870119903119900 (119878

119908

119894(119905)

119895) 119870119903119908

119894

119895= 119870119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895 119876

119905

119895=119879119895

119879sdot 119876119905

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895 119891

119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(2)

Here 119891(history)119908

(119905) is history water cut and 119891119908(119905) is cal-

culation water cut the first water cut is a known amountfrom production and another is calculated by our inverseproblem models Water cut means water production rate is akey parameter in reservoir engineering which can representwater production capacity in reservoir development If watercut is too high it may bring negative effects to reservoirproduction Therefore the history matching for water pro-duction rate plays an important role in reservoir dynamicanalysis and numerical simulation [4ndash6]

And thus 119894 = 1 2 3 119898 119895 = 1 2 3 119899 119898

is the average partition total 119899 is the partition total 119879is grid conductivity ℎ

119895is each longitudinal layer stratum

Journal of Applied Mathematics 3

dr

Supp

ly ed

ge

Fluid producing edge

h

re

Figure 2 Radial flow unit model

thickness 119870119895is each longitudinal layer permeability 120601

119895is

each longitudinal layer porosity 119878119894119908119895

is grid water saturationat different time 119878

119908119888is the initial water saturation 119864 is

the objective function 119903119890is reservoir radius 119870 is average

permeability 119870119903119900

is oil relative permeability 119870119903119908

is waterrelative permeability119876

119905is the total fluid production119876119905

119908is the

total water yield of fluid producing edge at different time 119876119905119900

is the total oil production of fluid producing edge at differenttime 119876119905

119895is the total fluid production of each longitudinal

layer at different time 119878119908119891

is the frontier saturation 119903119908119891

is thecorresponding displacement of frontier saturation

The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032

119890minus 1199032= 1198911015840

119908sdot 119876119905120601 sdot 120587 sdot ℎ

without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus

119891(history)119908

(119905))2 By the end an inverse problem model on the

optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water

relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on

the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies

2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow

Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (3)

Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2

According to the seepage principle [9] we can get a flowequation of the volume element

119902119908= 119902 sdot 119891

119908 (4)

In the 119889119905 time the flow volume of the volume element is

119876out = 119889119902119908sdot 119889119905 (5)

We can derive from (4) and (5) that

119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)

Meanwhile the inflow volume of the volume element is

119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)

In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows

119902 sdot119889119891119908

119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)

From 119891119908= 119891119908(119878119908) we can get that 1198911015840

119908= 119889119891119908119889119878119908

Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903

after infinitesimal calculus we obtain

1198911015840

119908int119905

0

119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890

119903

119903119889119903 (9)

If119876119905 = int119905

0119902 sdot119889119905 then (9) turns to1198911015840

119908sdot119876119905= 120601sdot120587ℎ sdot (119903

2

119890minus1199032)

We obtain

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (10)

From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(11)

Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from

the water cut function curve 119891119908

= 119891119908(119878119908) selecting the

point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891

119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878

119908119891) then

one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding

displacement of frontier saturation 119903119908119891

is

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(12)

4 Journal of Applied Mathematics

3 The Inverse Problem Model on Oil-WaterRelative Permeability

31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(13)

In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed

119870119903119908

(119878119894

119908119895) = 119886119908(

119878119894

119908119895minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119894

119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894

119908119895

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(14)

In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-

sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)

32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations

Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the

water cut 119891119908(119905) based on the model of the Theorem 1 if the

objective function 119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 can

be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886

119908 119887119908 119886119900 119887119900 and oil-

water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)

Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following

initial value 1198860

119908 1198870

119908 1198860

119900 1198870

119900

119870119903119908

(119878119908) = 1198860

119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119908

119870119903119900(119878119908) = 1198860

119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119900

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(15)

We can derive from (12) and (15) 119903119908119891

If 119903119890gt 119903119908119891 then 119878

119908

119894(119905)

119895= 119878119908119888119895

If 119903119890le 119903119908119891

solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908) and so forth 119894 = 1

If we obtain the value of the water saturation [12] relyingon the function 119891

119908(119878119908) sim 119878

119908 we can calculate the value

of the longitudinal each layer water cut 119891119905119908(1198781

119908119895) and the

constructing mathematic model as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895)

119876119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(16)

From (16) we can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and rely on the objective function 119864 =

minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 with numerical optimization

calculation in the inverse problem model an optimizationproblem of the distribution 119886

119908 119887119908 119886119900 119887119900 will be obtained

so the 119870119903119908(119878119908) and 119870

119903119900(119878119908) can be formed The proof of

Theorem 3 is completed

4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution

Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870

1 Ln119870

2 Ln119870

119899 always obey nor-

mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =

Ln119870 meanwhile one calculates the value of the 119891119908(119905) based

on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905

119905=1(119891119908(119905)minus119891

(history)119908

(119905))2 then the

inverse problem model (2) will have the optimal distributionLn119870

1 Ln119870

2 Ln119870

119899 and 119870

1 1198702 119870

119899

Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)

According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min

According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883

0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a

cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for

Journal of Applied Mathematics 5

probability curve then the area summation of every closedfigures 119878

119895= 119865(119883

119895) minus119865(119883

119895minus1) each longitudinal layer stratum

thickness ℎ119895= ℎ sdot 119878

119895 and 119883

119895= Δ119883 sdot 119895 + 119883

0 119870119895= 119890119883119895

if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878

119895=

119865(119883119895)minus119865(119883

119895minus1) 119878119895= Δ119878 = 1119899 ℎ

119895= ℎsdot119878

119895 and119883

119895= 119865minus1(119878119895)

119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of

vertical stratification)Relying on the distribution 119870

1 1198702 119870

119899 oil-water

relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =

119896119903119908(119878119908) [14] the conductivity definition of numerical reser-

voir simulation we get the following equations

119896119903119900119894

119895= 119896119903119900 (119878

119908

119894(119905)

119895) 119896119903119908

119894

119895= 119896119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(17)

We can derive from (12) and (17) that 119903119908119891119895

If 119903119890

le 119903119908119891119895

solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908

119894(119905)

119895= 119865minus1(119903119894

119895 120601119895 ℎ119895 119876119905

119895) and thus 119894 = 1 If 119903

119890gt 119903119908119891119895

then

119878119908

119894(119905)

119895= 119878119908119888119895

and thus 119894 = 1If we obtain the value of the water saturation (119878

119908) relying

on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the

longitudinal each layer water cut 119891119905119908(1198781

119908119895) then mathematic

model can be constructed as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

(119876119905119908+ 119876119905119900)

(18)

Equation (18) can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and relies on the objective function

119864 = minsum119899119905119905=1

(119891119908(119905)minus119891

(history)119908

(119905))2With numerical optimiza-

tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870

119899 will be obtained so the distribution

1198701 1198702 119870

119899 will be given The proof of Theorem 4 is

completed

5 Discussions and Conclusions

The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a

group of oil-water relative permeability equations 119870119903119908(119878119908)

and 119870119903119900(119878119908) Combining with the objective function

119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 through the optimization

solution method it will finally bring a set of the optimumdistribution 119886

119908 119887119908 119886119900 119887119900 and the oil-water relative

permeability equations119870119903119908(119878119908) and119870

119903119900(119878119908)

According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583

119894 1205902

119894) corresponds to its own normal distribution

curve with the certain expectation 120583119894 If 120590

119894goes up the

volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583

119894

each different value of 120590119894corresponds to a group of original

values of (120583119894 1205902

119894) which will yield a group of values of

1198701 1198702 119870

119899minus1 119870119899 namely the value of permeability of

each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus 119891

(history)119908

(119905))2

it will finally bring a set of optimum normal distributionsof 119883 sim (120583

119894 1205902

119894) (Etc 119883 = ln119870) through the optimization

solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model

Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance

Acknowledgments

The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions

References

[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012

[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012

[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013

[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011

[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

dr

Supp

ly ed

ge

Fluid producing edge

h

re

Figure 2 Radial flow unit model

thickness 119870119895is each longitudinal layer permeability 120601

119895is

each longitudinal layer porosity 119878119894119908119895

is grid water saturationat different time 119878

119908119888is the initial water saturation 119864 is

the objective function 119903119890is reservoir radius 119870 is average

permeability 119870119903119900

is oil relative permeability 119870119903119908

is waterrelative permeability119876

119905is the total fluid production119876119905

119908is the

total water yield of fluid producing edge at different time 119876119905119900

is the total oil production of fluid producing edge at differenttime 119876119905

119895is the total fluid production of each longitudinal

layer at different time 119878119908119891

is the frontier saturation 119903119908119891

is thecorresponding displacement of frontier saturation

The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032

119890minus 1199032= 1198911015840

119908sdot 119876119905120601 sdot 120587 sdot ℎ

without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus

119891(history)119908

(119905))2 By the end an inverse problem model on the

optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water

relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on

the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies

2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow

Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (3)

Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2

According to the seepage principle [9] we can get a flowequation of the volume element

119902119908= 119902 sdot 119891

119908 (4)

In the 119889119905 time the flow volume of the volume element is

119876out = 119889119902119908sdot 119889119905 (5)

We can derive from (4) and (5) that

119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)

Meanwhile the inflow volume of the volume element is

119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)

In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows

119902 sdot119889119891119908

119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)

From 119891119908= 119891119908(119878119908) we can get that 1198911015840

119908= 119889119891119908119889119878119908

Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903

after infinitesimal calculus we obtain

1198911015840

119908int119905

0

119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890

119903

119903119889119903 (9)

If119876119905 = int119905

0119902 sdot119889119905 then (9) turns to1198911015840

119908sdot119876119905= 120601sdot120587ℎ sdot (119903

2

119890minus1199032)

We obtain

1199032

119890minus 1199032=1198911015840

119908sdot 119876119905

120601 sdot 120587ℎ (10)

From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat

1199032

119890minus 119903119894

119895

2

=119891119905

119908(119878119908

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(11)

Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from

the water cut function curve 119891119908

= 119891119908(119878119908) selecting the

point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891

119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878

119908119891) then

one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding

displacement of frontier saturation 119903119908119891

is

119903119908119891

2

119895= 1199032

119890minus119891119905

119908(119878119908119891

119894

119895)1015840

119876119905

119895

120601119895sdot 120587ℎ119895

(12)

4 Journal of Applied Mathematics

3 The Inverse Problem Model on Oil-WaterRelative Permeability

31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(13)

In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed

119870119903119908

(119878119894

119908119895) = 119886119908(

119878119894

119908119895minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119894

119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894

119908119895

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(14)

In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-

sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)

32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations

Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the

water cut 119891119908(119905) based on the model of the Theorem 1 if the

objective function 119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 can

be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886

119908 119887119908 119886119900 119887119900 and oil-

water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)

Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following

initial value 1198860

119908 1198870

119908 1198860

119900 1198870

119900

119870119903119908

(119878119908) = 1198860

119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119908

119870119903119900(119878119908) = 1198860

119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119900

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(15)

We can derive from (12) and (15) 119903119908119891

If 119903119890gt 119903119908119891 then 119878

119908

119894(119905)

119895= 119878119908119888119895

If 119903119890le 119903119908119891

solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908) and so forth 119894 = 1

If we obtain the value of the water saturation [12] relyingon the function 119891

119908(119878119908) sim 119878

119908 we can calculate the value

of the longitudinal each layer water cut 119891119905119908(1198781

119908119895) and the

constructing mathematic model as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895)

119876119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(16)

From (16) we can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and rely on the objective function 119864 =

minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 with numerical optimization

calculation in the inverse problem model an optimizationproblem of the distribution 119886

119908 119887119908 119886119900 119887119900 will be obtained

so the 119870119903119908(119878119908) and 119870

119903119900(119878119908) can be formed The proof of

Theorem 3 is completed

4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution

Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870

1 Ln119870

2 Ln119870

119899 always obey nor-

mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =

Ln119870 meanwhile one calculates the value of the 119891119908(119905) based

on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905

119905=1(119891119908(119905)minus119891

(history)119908

(119905))2 then the

inverse problem model (2) will have the optimal distributionLn119870

1 Ln119870

2 Ln119870

119899 and 119870

1 1198702 119870

119899

Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)

According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min

According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883

0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a

cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for

Journal of Applied Mathematics 5

probability curve then the area summation of every closedfigures 119878

119895= 119865(119883

119895) minus119865(119883

119895minus1) each longitudinal layer stratum

thickness ℎ119895= ℎ sdot 119878

119895 and 119883

119895= Δ119883 sdot 119895 + 119883

0 119870119895= 119890119883119895

if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878

119895=

119865(119883119895)minus119865(119883

119895minus1) 119878119895= Δ119878 = 1119899 ℎ

119895= ℎsdot119878

119895 and119883

119895= 119865minus1(119878119895)

119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of

vertical stratification)Relying on the distribution 119870

1 1198702 119870

119899 oil-water

relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =

119896119903119908(119878119908) [14] the conductivity definition of numerical reser-

voir simulation we get the following equations

119896119903119900119894

119895= 119896119903119900 (119878

119908

119894(119905)

119895) 119896119903119908

119894

119895= 119896119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(17)

We can derive from (12) and (17) that 119903119908119891119895

If 119903119890

le 119903119908119891119895

solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908

119894(119905)

119895= 119865minus1(119903119894

119895 120601119895 ℎ119895 119876119905

119895) and thus 119894 = 1 If 119903

119890gt 119903119908119891119895

then

119878119908

119894(119905)

119895= 119878119908119888119895

and thus 119894 = 1If we obtain the value of the water saturation (119878

119908) relying

on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the

longitudinal each layer water cut 119891119905119908(1198781

119908119895) then mathematic

model can be constructed as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

(119876119905119908+ 119876119905119900)

(18)

Equation (18) can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and relies on the objective function

119864 = minsum119899119905119905=1

(119891119908(119905)minus119891

(history)119908

(119905))2With numerical optimiza-

tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870

119899 will be obtained so the distribution

1198701 1198702 119870

119899 will be given The proof of Theorem 4 is

completed

5 Discussions and Conclusions

The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a

group of oil-water relative permeability equations 119870119903119908(119878119908)

and 119870119903119900(119878119908) Combining with the objective function

119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 through the optimization

solution method it will finally bring a set of the optimumdistribution 119886

119908 119887119908 119886119900 119887119900 and the oil-water relative

permeability equations119870119903119908(119878119908) and119870

119903119900(119878119908)

According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583

119894 1205902

119894) corresponds to its own normal distribution

curve with the certain expectation 120583119894 If 120590

119894goes up the

volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583

119894

each different value of 120590119894corresponds to a group of original

values of (120583119894 1205902

119894) which will yield a group of values of

1198701 1198702 119870

119899minus1 119870119899 namely the value of permeability of

each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus 119891

(history)119908

(119905))2

it will finally bring a set of optimum normal distributionsof 119883 sim (120583

119894 1205902

119894) (Etc 119883 = ln119870) through the optimization

solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model

Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance

Acknowledgments

The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions

References

[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012

[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012

[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013

[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011

[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

3 The Inverse Problem Model on Oil-WaterRelative Permeability

31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows

119870119903119908

(119878119908) = 119886119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(13)

In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed

119870119903119908

(119878119894

119908119895) = 119886119908(

119878119894

119908119895minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

119887119908

119870119903119900(119878119894

119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894

119908119895

1 minus 119878119908119888

minus 119878119900119903

)

119887119900

(14)

In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-

sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)

32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations

Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the

water cut 119891119908(119905) based on the model of the Theorem 1 if the

objective function 119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 can

be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886

119908 119887119908 119886119900 119887119900 and oil-

water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)

Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following

initial value 1198860

119908 1198870

119908 1198860

119900 1198870

119900

119870119903119908

(119878119908) = 1198860

119908(

119878119908minus 119878119908119888

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119908

119870119903119900(119878119908) = 1198860

119900(1 minus 119878119900119903minus 119878119908

1 minus 119878119908119888

minus 119878119900119903

)

1198870

119900

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(15)

We can derive from (12) and (15) 119903119908119891

If 119903119890gt 119903119908119891 then 119878

119908

119894(119905)

119895= 119878119908119888119895

If 119903119890le 119903119908119891

solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878

119908

119894(119905)

119895= 119891minus1

119908

1015840

(119878119908) and so forth 119894 = 1

If we obtain the value of the water saturation [12] relyingon the function 119891

119908(119878119908) sim 119878

119908 we can calculate the value

of the longitudinal each layer water cut 119891119905119908(1198781

119908119895) and the

constructing mathematic model as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895)

119876119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

119876119905119908+ 119876119905119900

(16)

From (16) we can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and rely on the objective function 119864 =

minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 with numerical optimization

calculation in the inverse problem model an optimizationproblem of the distribution 119886

119908 119887119908 119886119900 119887119900 will be obtained

so the 119870119903119908(119878119908) and 119870

119903119900(119878119908) can be formed The proof of

Theorem 3 is completed

4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution

Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870

1 Ln119870

2 Ln119870

119899 always obey nor-

mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =

Ln119870 meanwhile one calculates the value of the 119891119908(119905) based

on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905

119905=1(119891119908(119905)minus119891

(history)119908

(119905))2 then the

inverse problem model (2) will have the optimal distributionLn119870

1 Ln119870

2 Ln119870

119899 and 119870

1 1198702 119870

119899

Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)

According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min

According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883

0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a

cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for

Journal of Applied Mathematics 5

probability curve then the area summation of every closedfigures 119878

119895= 119865(119883

119895) minus119865(119883

119895minus1) each longitudinal layer stratum

thickness ℎ119895= ℎ sdot 119878

119895 and 119883

119895= Δ119883 sdot 119895 + 119883

0 119870119895= 119890119883119895

if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878

119895=

119865(119883119895)minus119865(119883

119895minus1) 119878119895= Δ119878 = 1119899 ℎ

119895= ℎsdot119878

119895 and119883

119895= 119865minus1(119878119895)

119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of

vertical stratification)Relying on the distribution 119870

1 1198702 119870

119899 oil-water

relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =

119896119903119908(119878119908) [14] the conductivity definition of numerical reser-

voir simulation we get the following equations

119896119903119900119894

119895= 119896119903119900 (119878

119908

119894(119905)

119895) 119896119903119908

119894

119895= 119896119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(17)

We can derive from (12) and (17) that 119903119908119891119895

If 119903119890

le 119903119908119891119895

solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908

119894(119905)

119895= 119865minus1(119903119894

119895 120601119895 ℎ119895 119876119905

119895) and thus 119894 = 1 If 119903

119890gt 119903119908119891119895

then

119878119908

119894(119905)

119895= 119878119908119888119895

and thus 119894 = 1If we obtain the value of the water saturation (119878

119908) relying

on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the

longitudinal each layer water cut 119891119905119908(1198781

119908119895) then mathematic

model can be constructed as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

(119876119905119908+ 119876119905119900)

(18)

Equation (18) can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and relies on the objective function

119864 = minsum119899119905119905=1

(119891119908(119905)minus119891

(history)119908

(119905))2With numerical optimiza-

tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870

119899 will be obtained so the distribution

1198701 1198702 119870

119899 will be given The proof of Theorem 4 is

completed

5 Discussions and Conclusions

The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a

group of oil-water relative permeability equations 119870119903119908(119878119908)

and 119870119903119900(119878119908) Combining with the objective function

119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 through the optimization

solution method it will finally bring a set of the optimumdistribution 119886

119908 119887119908 119886119900 119887119900 and the oil-water relative

permeability equations119870119903119908(119878119908) and119870

119903119900(119878119908)

According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583

119894 1205902

119894) corresponds to its own normal distribution

curve with the certain expectation 120583119894 If 120590

119894goes up the

volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583

119894

each different value of 120590119894corresponds to a group of original

values of (120583119894 1205902

119894) which will yield a group of values of

1198701 1198702 119870

119899minus1 119870119899 namely the value of permeability of

each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus 119891

(history)119908

(119905))2

it will finally bring a set of optimum normal distributionsof 119883 sim (120583

119894 1205902

119894) (Etc 119883 = ln119870) through the optimization

solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model

Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance

Acknowledgments

The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions

References

[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012

[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012

[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013

[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011

[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

probability curve then the area summation of every closedfigures 119878

119895= 119865(119883

119895) minus119865(119883

119895minus1) each longitudinal layer stratum

thickness ℎ119895= ℎ sdot 119878

119895 and 119883

119895= Δ119883 sdot 119895 + 119883

0 119870119895= 119890119883119895

if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878

119895=

119865(119883119895)minus119865(119883

119895minus1) 119878119895= Δ119878 = 1119899 ℎ

119895= ℎsdot119878

119895 and119883

119895= 119865minus1(119878119895)

119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of

vertical stratification)Relying on the distribution 119870

1 1198702 119870

119899 oil-water

relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =

119896119903119908(119878119908) [14] the conductivity definition of numerical reser-

voir simulation we get the following equations

119896119903119900119894

119895= 119896119903119900 (119878

119908

119894(119905)

119895) 119896119903119908

119894

119895= 119896119903119908 (119878

119908

119894(119905)

119895)

119879119894

119895=

2120587ℎ119895119870119895

ln (119903119903119894119895)(119870119903119900

119894

119895

120583119900

+119870119903119908

119894

119895

120583119908

)

119879119895=

1

sum119898

119894=1(1119879119894119895) 119879 =

119899

sum119895=1

119879119895

119876119905

119895=119879119895

119879sdot 119876119905

(17)

We can derive from (12) and (17) that 119903119908119891119895

If 119903119890

le 119903119908119891119895

solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908

119894(119905)

119895= 119865minus1(119903119894

119895 120601119895 ℎ119895 119876119905

119895) and thus 119894 = 1 If 119903

119890gt 119903119908119891119895

then

119878119908

119894(119905)

119895= 119878119908119888119895

and thus 119894 = 1If we obtain the value of the water saturation (119878

119908) relying

on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the

longitudinal each layer water cut 119891119905119908(1198781

119908119895) then mathematic

model can be constructed as follows

119876119905

119908119895= 119876119905

119895sdot 119891119905

119908(1198781

119908119895) 119876

119905

119900119895= 119876119905

119895sdot (1 minus 119891

119905

119908(1198781

119908119895))

119876119905

119908=

119899

sum119895=1

119876119905

119908119895 119876

119905

119900=

119899

sum119895=1

119876119905

119900119895

119891119908(119905) =

119876119905

119908

(119876119905119908+ 119876119905119900)

(18)

Equation (18) can calculate the total water cut of the fluidproducing edge 119891

119908(119905) and relies on the objective function

119864 = minsum119899119905119905=1

(119891119908(119905)minus119891

(history)119908

(119905))2With numerical optimiza-

tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870

119899 will be obtained so the distribution

1198701 1198702 119870

119899 will be given The proof of Theorem 4 is

completed

5 Discussions and Conclusions

The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a

group of oil-water relative permeability equations 119870119903119908(119878119908)

and 119870119903119900(119878119908) Combining with the objective function

119864 = minsum119899119905119905=1

(119891119908(119905) minus 119891

(history)119908

(119905))2 through the optimization

solution method it will finally bring a set of the optimumdistribution 119886

119908 119887119908 119886119900 119887119900 and the oil-water relative

permeability equations119870119903119908(119878119908) and119870

119903119900(119878119908)

According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583

119894 1205902

119894) corresponds to its own normal distribution

curve with the certain expectation 120583119894 If 120590

119894goes up the

volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583

119894

each different value of 120590119894corresponds to a group of original

values of (120583119894 1205902

119894) which will yield a group of values of

1198701 1198702 119870

119899minus1 119870119899 namely the value of permeability of

each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905

119905=1(119891119908(119905) minus 119891

(history)119908

(119905))2

it will finally bring a set of optimum normal distributionsof 119883 sim (120583

119894 1205902

119894) (Etc 119883 = ln119870) through the optimization

solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model

Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance

Acknowledgments

The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions

References

[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012

[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012

[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013

[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011

[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011

[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003

[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010

[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998

[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012

[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006

[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012

[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of